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Theorem dffi3 8220
 Description: The set of finite intersections can be "constructed" inductively by iterating binary intersection ω-many times. (Contributed by Mario Carneiro, 21-Mar-2015.)
Hypothesis
Ref Expression
dffi3.1 𝑅 = (𝑢 ∈ V ↦ ran (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)))
Assertion
Ref Expression
dffi3 (𝐴𝑉 → (fi‘𝐴) = (rec(𝑅, 𝐴) “ ω))
Distinct variable groups:   𝑦,𝐴   𝑦,𝑅   𝑦,𝑉   𝑦,𝑢,𝑧
Allowed substitution hints:   𝐴(𝑧,𝑢)   𝑅(𝑧,𝑢)   𝑉(𝑧,𝑢)

Proof of Theorem dffi3
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑚 𝑛 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dffi2 8212 . . . 4 (𝐴𝑉 → (fi‘𝐴) = {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)})
2 fr0g 7418 . . . . . . . 8 (𝐴𝑉 → ((rec(𝑅, 𝐴) ↾ ω)‘∅) = 𝐴)
3 frfnom 7417 . . . . . . . . 9 (rec(𝑅, 𝐴) ↾ ω) Fn ω
4 peano1 6977 . . . . . . . . 9 ∅ ∈ ω
5 fnfvelrn 6264 . . . . . . . . 9 (((rec(𝑅, 𝐴) ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((rec(𝑅, 𝐴) ↾ ω)‘∅) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
63, 4, 5mp2an 704 . . . . . . . 8 ((rec(𝑅, 𝐴) ↾ ω)‘∅) ∈ ran (rec(𝑅, 𝐴) ↾ ω)
72, 6syl6eqelr 2697 . . . . . . 7 (𝐴𝑉𝐴 ∈ ran (rec(𝑅, 𝐴) ↾ ω))
8 elssuni 4403 . . . . . . 7 (𝐴 ∈ ran (rec(𝑅, 𝐴) ↾ ω) → 𝐴 ran (rec(𝑅, 𝐴) ↾ ω))
97, 8syl 17 . . . . . 6 (𝐴𝑉𝐴 ran (rec(𝑅, 𝐴) ↾ ω))
10 reeanv 3086 . . . . . . . . 9 (∃𝑚 ∈ ω ∃𝑛 ∈ ω (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) ↔ (∃𝑚 ∈ ω 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ ∃𝑛 ∈ ω 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)))
11 eliun 4460 . . . . . . . . . 10 (𝑐 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ↔ ∃𝑚 ∈ ω 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚))
12 eliun 4460 . . . . . . . . . 10 (𝑑 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ ∃𝑛 ∈ ω 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
1311, 12anbi12i 729 . . . . . . . . 9 ((𝑐 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) ↔ (∃𝑚 ∈ ω 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ ∃𝑛 ∈ ω 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)))
14 fniunfv 6409 . . . . . . . . . . . 12 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) = ran (rec(𝑅, 𝐴) ↾ ω))
1514eleq2d 2673 . . . . . . . . . . 11 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → (𝑐 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ↔ 𝑐 ran (rec(𝑅, 𝐴) ↾ ω)))
16 fniunfv 6409 . . . . . . . . . . . 12 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) = ran (rec(𝑅, 𝐴) ↾ ω))
1716eleq2d 2673 . . . . . . . . . . 11 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → (𝑑 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ 𝑑 ran (rec(𝑅, 𝐴) ↾ ω)))
1815, 17anbi12d 743 . . . . . . . . . 10 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → ((𝑐 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) ↔ (𝑐 ran (rec(𝑅, 𝐴) ↾ ω) ∧ 𝑑 ran (rec(𝑅, 𝐴) ↾ ω))))
193, 18ax-mp 5 . . . . . . . . 9 ((𝑐 𝑚 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 𝑛 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) ↔ (𝑐 ran (rec(𝑅, 𝐴) ↾ ω) ∧ 𝑑 ran (rec(𝑅, 𝐴) ↾ ω)))
2010, 13, 193bitr2i 287 . . . . . . . 8 (∃𝑚 ∈ ω ∃𝑛 ∈ ω (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) ↔ (𝑐 ran (rec(𝑅, 𝐴) ↾ ω) ∧ 𝑑 ran (rec(𝑅, 𝐴) ↾ ω)))
21 ordom 6966 . . . . . . . . . . . . . . . 16 Ord ω
22 ordunel 6919 . . . . . . . . . . . . . . . 16 ((Ord ω ∧ 𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (𝑚𝑛) ∈ ω)
2321, 22mp3an1 1403 . . . . . . . . . . . . . . 15 ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (𝑚𝑛) ∈ ω)
2423adantl 481 . . . . . . . . . . . . . 14 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → (𝑚𝑛) ∈ ω)
25 simprl 790 . . . . . . . . . . . . . 14 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑚 ∈ ω)
2624, 25jca 553 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑚𝑛) ∈ ω ∧ 𝑚 ∈ ω))
27 nnon 6963 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ω → 𝑦 ∈ On)
2827adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → 𝑦 ∈ On)
29 nnon 6963 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ω → 𝑥 ∈ On)
3029ad2antlr 759 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → 𝑥 ∈ On)
31 onsseleq 5682 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦𝑥 ↔ (𝑦𝑥𝑦 = 𝑥)))
3228, 30, 31syl2anc 691 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦𝑥 ↔ (𝑦𝑥𝑦 = 𝑥)))
33 rzal 4025 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = ∅ → ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
3433biantrud 527 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ∅ → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))))
35 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = ∅ → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘∅))
3635sseq1d 3595 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ∅ → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘∅) ⊆ (fi‘𝐴)))
3734, 36bitr3d 269 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = ∅ → ((((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘∅) ⊆ (fi‘𝐴)))
38 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝑛 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
3938sseq1d 3595 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)))
4038sseq2d 3596 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)))
4140raleqbi1dv 3123 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑛 → (∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)))
4239, 41anbi12d 743 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑛 → ((((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) ↔ (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))))
43 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = suc 𝑛 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
4443sseq1d 3595 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = suc 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴)))
4543sseq2d 3596 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = suc 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
4645raleqbi1dv 3123 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = suc 𝑛 → (∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
4744, 46anbi12d 743 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = suc 𝑛 → ((((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) ↔ (((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))))
48 ssfii 8208 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴𝑉𝐴 ⊆ (fi‘𝐴))
492, 48eqsstrd 3602 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴𝑉 → ((rec(𝑅, 𝐴) ↾ ω)‘∅) ⊆ (fi‘𝐴))
50 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
51 eqidd 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑥 = 𝑥)
52 ineq1 3769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑎 = 𝑥 → (𝑎𝑏) = (𝑥𝑏))
5352eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑎 = 𝑥 → (𝑥 = (𝑎𝑏) ↔ 𝑥 = (𝑥𝑏)))
54 ineq2 3770 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑏 = 𝑥 → (𝑥𝑏) = (𝑥𝑥))
55 inidm 3784 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑥𝑥) = 𝑥
5654, 55syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑏 = 𝑥 → (𝑥𝑏) = 𝑥)
5756eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑏 = 𝑥 → (𝑥 = (𝑥𝑏) ↔ 𝑥 = 𝑥))
5853, 57rspc2ev 3295 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑥 = 𝑥) → ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)𝑥 = (𝑎𝑏))
5950, 50, 51, 58syl3anc 1318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)𝑥 = (𝑎𝑏))
60 eqid 2610 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏))
6160rnmpt2 6668 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) = {𝑥 ∣ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)𝑥 = (𝑎𝑏)}
6261abeq2i 2722 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑥 ∈ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ↔ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)𝑥 = (𝑎𝑏))
6359, 62sylibr 223 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑥 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑥 ∈ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)))
6463ssriv 3572 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏))
65 simpl 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → 𝑛 ∈ ω)
66 fvex 6113 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∈ V
6766uniex 6851 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∈ V
6867pwex 4774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∈ V
69 inss1 3795 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑎𝑏) ⊆ 𝑎
70 elssuni 4403 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑎 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7170adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → 𝑎 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7269, 71syl5ss 3579 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑎𝑏) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
73 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 𝑎 ∈ V
7473inex1 4727 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑎𝑏) ∈ V
7574elpw 4114 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ (𝑎𝑏) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7672, 75sylibr 223 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7776rgen2a 2960 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)
7860fmpt2 7126 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↔ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘𝑛) × ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))⟶𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
7977, 78mpbi 219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘𝑛) × ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))⟶𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)
80 frn 5966 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘𝑛) × ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))⟶𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ⊆ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
8179, 80ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ⊆ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)
8268, 81ssexi 4731 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ∈ V
83 nfcv 2751 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝑣𝐴
84 nfcv 2751 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝑣𝑛
85 nfcv 2751 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝑣ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏))
86 dffi3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 𝑅 = (𝑢 ∈ V ↦ ran (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)))
87 mpt2eq12 6613 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝑢 = 𝑣𝑢 = 𝑣) → (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)) = (𝑦𝑣, 𝑧𝑣 ↦ (𝑦𝑧)))
8887anidms 675 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑢 = 𝑣 → (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)) = (𝑦𝑣, 𝑧𝑣 ↦ (𝑦𝑧)))
89 ineq1 3769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑦 = 𝑎 → (𝑦𝑧) = (𝑎𝑧))
90 ineq2 3770 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑧 = 𝑏 → (𝑎𝑧) = (𝑎𝑏))
9189, 90cbvmpt2v 6633 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑦𝑣, 𝑧𝑣 ↦ (𝑦𝑧)) = (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏))
9288, 91syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑢 = 𝑣 → (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)) = (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)))
9392rneqd 5274 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑢 = 𝑣 → ran (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧)) = ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)))
9493cbvmptv 4678 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑢 ∈ V ↦ ran (𝑦𝑢, 𝑧𝑢 ↦ (𝑦𝑧))) = (𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)))
9586, 94eqtri 2632 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 𝑅 = (𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)))
96 rdgeq1 7394 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑅 = (𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏))) → rec(𝑅, 𝐴) = rec((𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏))), 𝐴))
9795, 96ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 rec(𝑅, 𝐴) = rec((𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏))), 𝐴)
9897reseq1i 5313 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (rec(𝑅, 𝐴) ↾ ω) = (rec((𝑣 ∈ V ↦ ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏))), 𝐴) ↾ ω)
99 mpt2eq12 6613 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ∧ 𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)))
10099anidms 675 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)))
101100rneqd 5274 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)))
10283, 84, 85, 98, 101frsucmpt 7420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑛 ∈ ω ∧ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ∈ V) → ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)))
10365, 82, 102sylancl 693 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)))
10464, 103syl5sseqr 3617 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
105 sstr2 3575 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
106104, 105syl5com 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
107106ralimdv 2946 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → (∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
108 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 𝑛 ∈ V
109 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = 𝑛 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))
110109sseq1d 3595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
111108, 110ralsn 4169 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∀𝑦 ∈ {𝑛} ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
112104, 111sylibr 223 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ∀𝑦 ∈ {𝑛} ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
113107, 112jctird 565 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → (∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → (∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ∧ ∀𝑦 ∈ {𝑛} ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))))
114 df-suc 5646 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 suc 𝑛 = (𝑛 ∪ {𝑛})
115114raleqi 3119 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ↔ ∀𝑦 ∈ (𝑛 ∪ {𝑛})((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))
116 ralunb 3756 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∀𝑦 ∈ (𝑛 ∪ {𝑛})((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ↔ (∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ∧ ∀𝑦 ∈ {𝑛} ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
117115, 116bitri 263 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ↔ (∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ∧ ∀𝑦 ∈ {𝑛} ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
118113, 117syl6ibr 241 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → (∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))
119 fiin 8211 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑎 ∈ (fi‘𝐴) ∧ 𝑏 ∈ (fi‘𝐴)) → (𝑎𝑏) ∈ (fi‘𝐴))
120119rgen2a 2960 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑎 ∈ (fi‘𝐴)∀𝑏 ∈ (fi‘𝐴)(𝑎𝑏) ∈ (fi‘𝐴)
121 ssralv 3629 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → (∀𝑏 ∈ (fi‘𝐴)(𝑎𝑏) ∈ (fi‘𝐴) → ∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ (fi‘𝐴)))
122121ralimdv 2946 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → (∀𝑎 ∈ (fi‘𝐴)∀𝑏 ∈ (fi‘𝐴)(𝑎𝑏) ∈ (fi‘𝐴) → ∀𝑎 ∈ (fi‘𝐴)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ (fi‘𝐴)))
123 ssralv 3629 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → (∀𝑎 ∈ (fi‘𝐴)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ (fi‘𝐴) → ∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ (fi‘𝐴)))
124122, 123syld 46 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → (∀𝑎 ∈ (fi‘𝐴)∀𝑏 ∈ (fi‘𝐴)(𝑎𝑏) ∈ (fi‘𝐴) → ∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ (fi‘𝐴)))
125120, 124mpi 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → ∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ (fi‘𝐴))
12660fmpt2 7126 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)(𝑎𝑏) ∈ (fi‘𝐴) ↔ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘𝑛) × ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))⟶(fi‘𝐴))
127125, 126sylib 207 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘𝑛) × ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))⟶(fi‘𝐴))
128 frn 5966 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘𝑛) × ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))⟶(fi‘𝐴) → ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ⊆ (fi‘𝐴))
129127, 128syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) → ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ⊆ (fi‘𝐴))
130129adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ↦ (𝑎𝑏)) ⊆ (fi‘𝐴))
131103, 130eqsstrd 3602 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴))
132118, 131jctild 564 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑛 ∈ ω ∧ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴)) → (∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → (((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))))
133132expimpd 627 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ω → ((((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛))))
134133a1d 25 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ ω → (𝐴𝑉 → ((((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑛 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛) ⊆ (fi‘𝐴) ∧ ∀𝑦 ∈ suc 𝑛((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘suc 𝑛)))))
13537, 42, 47, 49, 134finds2 6986 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ ω → (𝐴𝑉 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))))
136135impcom 445 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝑉𝑥 ∈ ω) → (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴) ∧ ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
137136simprd 478 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴𝑉𝑥 ∈ ω) → ∀𝑦𝑥 ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
138137r19.21bi 2916 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦𝑥) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
139138ex 449 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑉𝑥 ∈ ω) → (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
140139adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
141 fveq2 6103 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
142 eqimss 3620 . . . . . . . . . . . . . . . . . . . 20 (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
143141, 142syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
144143a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦 = 𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
145140, 144jaod 394 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → ((𝑦𝑥𝑦 = 𝑥) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
14632, 145sylbid 229 . . . . . . . . . . . . . . . 16 (((𝐴𝑉𝑥 ∈ ω) ∧ 𝑦 ∈ ω) → (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
147146ralrimiva 2949 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝑥 ∈ ω) → ∀𝑦 ∈ ω (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
148147ralrimiva 2949 . . . . . . . . . . . . . 14 (𝐴𝑉 → ∀𝑥 ∈ ω ∀𝑦 ∈ ω (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
149148adantr 480 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ∀𝑥 ∈ ω ∀𝑦 ∈ ω (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)))
150 ssun1 3738 . . . . . . . . . . . . . 14 𝑚 ⊆ (𝑚𝑛)
151150a1i 11 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑚 ⊆ (𝑚𝑛))
152 sseq2 3590 . . . . . . . . . . . . . . 15 (𝑥 = (𝑚𝑛) → (𝑦𝑥𝑦 ⊆ (𝑚𝑛)))
153 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
154153sseq2d 3596 . . . . . . . . . . . . . . 15 (𝑥 = (𝑚𝑛) → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))))
155152, 154imbi12d 333 . . . . . . . . . . . . . 14 (𝑥 = (𝑚𝑛) → ((𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) ↔ (𝑦 ⊆ (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))))
156 sseq1 3589 . . . . . . . . . . . . . . 15 (𝑦 = 𝑚 → (𝑦 ⊆ (𝑚𝑛) ↔ 𝑚 ⊆ (𝑚𝑛)))
157 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑚 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) = ((rec(𝑅, 𝐴) ↾ ω)‘𝑚))
158157sseq1d 3595 . . . . . . . . . . . . . . 15 (𝑦 = 𝑚 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))))
159156, 158imbi12d 333 . . . . . . . . . . . . . 14 (𝑦 = 𝑚 → ((𝑦 ⊆ (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))) ↔ (𝑚 ⊆ (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))))
160155, 159rspc2v 3293 . . . . . . . . . . . . 13 (((𝑚𝑛) ∈ ω ∧ 𝑚 ∈ ω) → (∀𝑥 ∈ ω ∀𝑦 ∈ ω (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) → (𝑚 ⊆ (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))))
16126, 149, 151, 160syl3c 64 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
162161sseld 3567 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) → 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))))
163 simprr 792 . . . . . . . . . . . . . 14 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑛 ∈ ω)
16424, 163jca 553 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑚𝑛) ∈ ω ∧ 𝑛 ∈ ω))
165 ssun2 3739 . . . . . . . . . . . . . 14 𝑛 ⊆ (𝑚𝑛)
166165a1i 11 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → 𝑛 ⊆ (𝑚𝑛))
167 sseq1 3589 . . . . . . . . . . . . . . 15 (𝑦 = 𝑛 → (𝑦 ⊆ (𝑚𝑛) ↔ 𝑛 ⊆ (𝑚𝑛)))
168109sseq1d 3595 . . . . . . . . . . . . . . 15 (𝑦 = 𝑛 → (((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))))
169167, 168imbi12d 333 . . . . . . . . . . . . . 14 (𝑦 = 𝑛 → ((𝑦 ⊆ (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))) ↔ (𝑛 ⊆ (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))))
170155, 169rspc2v 3293 . . . . . . . . . . . . 13 (((𝑚𝑛) ∈ ω ∧ 𝑛 ∈ ω) → (∀𝑥 ∈ ω ∀𝑦 ∈ ω (𝑦𝑥 → ((rec(𝑅, 𝐴) ↾ ω)‘𝑦) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥)) → (𝑛 ⊆ (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))))
171164, 149, 166, 170syl3c 64 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
172171sseld 3567 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → (𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛) → 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))))
17323ad2antlr 759 . . . . . . . . . . . . . . 15 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑚𝑛) ∈ ω)
174 peano2 6978 . . . . . . . . . . . . . . 15 ((𝑚𝑛) ∈ ω → suc (𝑚𝑛) ∈ ω)
175 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑥 = suc (𝑚𝑛) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚𝑛)))
176175ssiun2s 4500 . . . . . . . . . . . . . . 15 (suc (𝑚𝑛) ∈ ω → ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚𝑛)) ⊆ 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
177173, 174, 1763syl 18 . . . . . . . . . . . . . 14 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚𝑛)) ⊆ 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
178 simprl 790 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → 𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
179 simprr 792 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
180 eqidd 2611 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑐𝑑) = (𝑐𝑑))
181 ineq1 3769 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝑐 → (𝑎𝑏) = (𝑐𝑏))
182181eqeq2d 2620 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑐 → ((𝑐𝑑) = (𝑎𝑏) ↔ (𝑐𝑑) = (𝑐𝑏)))
183 ineq2 3770 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = 𝑑 → (𝑐𝑏) = (𝑐𝑑))
184183eqeq2d 2620 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑑 → ((𝑐𝑑) = (𝑐𝑏) ↔ (𝑐𝑑) = (𝑐𝑑)))
185182, 184rspc2ev 3295 . . . . . . . . . . . . . . . . . 18 ((𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ (𝑐𝑑) = (𝑐𝑑)) → ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))(𝑐𝑑) = (𝑎𝑏))
186178, 179, 180, 185syl3anc 1318 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))(𝑐𝑑) = (𝑎𝑏))
187 vex 3176 . . . . . . . . . . . . . . . . . . 19 𝑐 ∈ V
188187inex1 4727 . . . . . . . . . . . . . . . . . 18 (𝑐𝑑) ∈ V
189 eqeq1 2614 . . . . . . . . . . . . . . . . . . 19 (𝑥 = (𝑐𝑑) → (𝑥 = (𝑎𝑏) ↔ (𝑐𝑑) = (𝑎𝑏)))
1901892rexbidv 3039 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑐𝑑) → (∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))𝑥 = (𝑎𝑏) ↔ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))(𝑐𝑑) = (𝑎𝑏)))
191188, 190elab 3319 . . . . . . . . . . . . . . . . 17 ((𝑐𝑑) ∈ {𝑥 ∣ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))𝑥 = (𝑎𝑏)} ↔ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))(𝑐𝑑) = (𝑎𝑏))
192186, 191sylibr 223 . . . . . . . . . . . . . . . 16 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑐𝑑) ∈ {𝑥 ∣ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))𝑥 = (𝑎𝑏)})
193 eqid 2610 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏))
194193rnmpt2 6668 . . . . . . . . . . . . . . . 16 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)) = {𝑥 ∣ ∃𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∃𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))𝑥 = (𝑎𝑏)}
195192, 194syl6eleqr 2699 . . . . . . . . . . . . . . 15 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑐𝑑) ∈ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)))
196 fvex 6113 . . . . . . . . . . . . . . . . . . 19 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∈ V
197196uniex 6851 . . . . . . . . . . . . . . . . . 18 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∈ V
198197pwex 4774 . . . . . . . . . . . . . . . . 17 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∈ V
199 elssuni 4403 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → 𝑎 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
20069, 199syl5ss 3579 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → (𝑎𝑏) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
20174elpw 4114 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↔ (𝑎𝑏) ⊆ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
202200, 201sylibr 223 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → (𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
203202adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))) → (𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
204203rgen2a 2960 . . . . . . . . . . . . . . . . . . 19 𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))(𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))
205193fmpt2 7126 . . . . . . . . . . . . . . . . . . 19 (∀𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))∀𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))(𝑎𝑏) ∈ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↔ (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) × ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))⟶𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
206204, 205mpbi 219 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) × ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))⟶𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))
207 frn 5966 . . . . . . . . . . . . . . . . . 18 ((𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)):(((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) × ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))⟶𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)) ⊆ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))
208206, 207ax-mp 5 . . . . . . . . . . . . . . . . 17 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)) ⊆ 𝒫 ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))
209198, 208ssexi 4731 . . . . . . . . . . . . . . . 16 ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)) ∈ V
210 nfcv 2751 . . . . . . . . . . . . . . . . 17 𝑣(𝑚𝑛)
211 nfcv 2751 . . . . . . . . . . . . . . . . 17 𝑣ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏))
212 mpt2eq12 6613 . . . . . . . . . . . . . . . . . . 19 ((𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))) → (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)))
213212anidms 675 . . . . . . . . . . . . . . . . . 18 (𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)) = (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)))
214213rneqd 5274 . . . . . . . . . . . . . . . . 17 (𝑣 = ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) → ran (𝑎𝑣, 𝑏𝑣 ↦ (𝑎𝑏)) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)))
21583, 210, 211, 98, 214frsucmpt 7420 . . . . . . . . . . . . . . . 16 (((𝑚𝑛) ∈ ω ∧ ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)) ∈ V) → ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚𝑛)) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)))
216173, 209, 215sylancl 693 . . . . . . . . . . . . . . 15 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚𝑛)) = ran (𝑎 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)), 𝑏 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ↦ (𝑎𝑏)))
217195, 216eleqtrrd 2691 . . . . . . . . . . . . . 14 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑐𝑑) ∈ ((rec(𝑅, 𝐴) ↾ ω)‘suc (𝑚𝑛)))
218177, 217sseldd 3569 . . . . . . . . . . . . 13 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑐𝑑) ∈ 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥))
219 fniunfv 6409 . . . . . . . . . . . . . 14 ((rec(𝑅, 𝐴) ↾ ω) Fn ω → 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ran (rec(𝑅, 𝐴) ↾ ω))
2203, 219ax-mp 5 . . . . . . . . . . . . 13 𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) = ran (rec(𝑅, 𝐴) ↾ ω)
221218, 220syl6eleq 2698 . . . . . . . . . . . 12 (((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) ∧ (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)))) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
222221ex 449 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛)) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘(𝑚𝑛))) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
223162, 172, 222syl2and 499 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑚 ∈ ω ∧ 𝑛 ∈ ω)) → ((𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
224223rexlimdvva 3020 . . . . . . . . 9 (𝐴𝑉 → (∃𝑚 ∈ ω ∃𝑛 ∈ ω (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛)) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
225224imp 444 . . . . . . . 8 ((𝐴𝑉 ∧ ∃𝑚 ∈ ω ∃𝑛 ∈ ω (𝑐 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑚) ∧ 𝑑 ∈ ((rec(𝑅, 𝐴) ↾ ω)‘𝑛))) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
22620, 225sylan2br 492 . . . . . . 7 ((𝐴𝑉 ∧ (𝑐 ran (rec(𝑅, 𝐴) ↾ ω) ∧ 𝑑 ran (rec(𝑅, 𝐴) ↾ ω))) → (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
227226ralrimivva 2954 . . . . . 6 (𝐴𝑉 → ∀𝑐 ran (rec(𝑅, 𝐴) ↾ ω)∀𝑑 ran (rec(𝑅, 𝐴) ↾ ω)(𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))
228136simpld 474 . . . . . . . . . . . 12 ((𝐴𝑉𝑥 ∈ ω) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴))
229 fvex 6113 . . . . . . . . . . . . 13 (fi‘𝐴) ∈ V
230229elpw2 4755 . . . . . . . . . . . 12 (((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴) ↔ ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ⊆ (fi‘𝐴))
231228, 230sylibr 223 . . . . . . . . . . 11 ((𝐴𝑉𝑥 ∈ ω) → ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴))
232231ralrimiva 2949 . . . . . . . . . 10 (𝐴𝑉 → ∀𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴))
233 fnfvrnss 6297 . . . . . . . . . 10 (((rec(𝑅, 𝐴) ↾ ω) Fn ω ∧ ∀𝑥 ∈ ω ((rec(𝑅, 𝐴) ↾ ω)‘𝑥) ∈ 𝒫 (fi‘𝐴)) → ran (rec(𝑅, 𝐴) ↾ ω) ⊆ 𝒫 (fi‘𝐴))
2343, 232, 233sylancr 694 . . . . . . . . 9 (𝐴𝑉 → ran (rec(𝑅, 𝐴) ↾ ω) ⊆ 𝒫 (fi‘𝐴))
235 sspwuni 4547 . . . . . . . . 9 (ran (rec(𝑅, 𝐴) ↾ ω) ⊆ 𝒫 (fi‘𝐴) ↔ ran (rec(𝑅, 𝐴) ↾ ω) ⊆ (fi‘𝐴))
236234, 235sylib 207 . . . . . . . 8 (𝐴𝑉 ran (rec(𝑅, 𝐴) ↾ ω) ⊆ (fi‘𝐴))
237 ssexg 4732 . . . . . . . 8 (( ran (rec(𝑅, 𝐴) ↾ ω) ⊆ (fi‘𝐴) ∧ (fi‘𝐴) ∈ V) → ran (rec(𝑅, 𝐴) ↾ ω) ∈ V)
238236, 229, 237sylancl 693 . . . . . . 7 (𝐴𝑉 ran (rec(𝑅, 𝐴) ↾ ω) ∈ V)
239 sseq2 3590 . . . . . . . . 9 (𝑥 = ran (rec(𝑅, 𝐴) ↾ ω) → (𝐴𝑥𝐴 ran (rec(𝑅, 𝐴) ↾ ω)))
240 eleq2 2677 . . . . . . . . . . 11 (𝑥 = ran (rec(𝑅, 𝐴) ↾ ω) → ((𝑐𝑑) ∈ 𝑥 ↔ (𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
241240raleqbi1dv 3123 . . . . . . . . . 10 (𝑥 = ran (rec(𝑅, 𝐴) ↾ ω) → (∀𝑑𝑥 (𝑐𝑑) ∈ 𝑥 ↔ ∀𝑑 ran (rec(𝑅, 𝐴) ↾ ω)(𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
242241raleqbi1dv 3123 . . . . . . . . 9 (𝑥 = ran (rec(𝑅, 𝐴) ↾ ω) → (∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥 ↔ ∀𝑐 ran (rec(𝑅, 𝐴) ↾ ω)∀𝑑 ran (rec(𝑅, 𝐴) ↾ ω)(𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω)))
243239, 242anbi12d 743 . . . . . . . 8 (𝑥 = ran (rec(𝑅, 𝐴) ↾ ω) → ((𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥) ↔ (𝐴 ran (rec(𝑅, 𝐴) ↾ ω) ∧ ∀𝑐 ran (rec(𝑅, 𝐴) ↾ ω)∀𝑑 ran (rec(𝑅, 𝐴) ↾ ω)(𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))))
244243elabg 3320 . . . . . . 7 ( ran (rec(𝑅, 𝐴) ↾ ω) ∈ V → ( ran (rec(𝑅, 𝐴) ↾ ω) ∈ {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)} ↔ (𝐴 ran (rec(𝑅, 𝐴) ↾ ω) ∧ ∀𝑐 ran (rec(𝑅, 𝐴) ↾ ω)∀𝑑 ran (rec(𝑅, 𝐴) ↾ ω)(𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))))
245238, 244syl 17 . . . . . 6 (𝐴𝑉 → ( ran (rec(𝑅, 𝐴) ↾ ω) ∈ {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)} ↔ (𝐴 ran (rec(𝑅, 𝐴) ↾ ω) ∧ ∀𝑐 ran (rec(𝑅, 𝐴) ↾ ω)∀𝑑 ran (rec(𝑅, 𝐴) ↾ ω)(𝑐𝑑) ∈ ran (rec(𝑅, 𝐴) ↾ ω))))
2469, 227, 245mpbir2and 959 . . . . 5 (𝐴𝑉 ran (rec(𝑅, 𝐴) ↾ ω) ∈ {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)})
247 intss1 4427 . . . . 5 ( ran (rec(𝑅, 𝐴) ↾ ω) ∈ {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)} → {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)} ⊆ ran (rec(𝑅, 𝐴) ↾ ω))
248246, 247syl 17 . . . 4 (𝐴𝑉 {𝑥 ∣ (𝐴𝑥 ∧ ∀𝑐𝑥𝑑𝑥 (𝑐𝑑) ∈ 𝑥)} ⊆ ran (rec(𝑅, 𝐴) ↾ ω))
2491, 248eqsstrd 3602 . . 3 (𝐴𝑉 → (fi‘𝐴) ⊆ ran (rec(𝑅, 𝐴) ↾ ω))
250249, 236eqssd 3585 . 2 (𝐴𝑉 → (fi‘𝐴) = ran (rec(𝑅, 𝐴) ↾ ω))
251 df-ima 5051 . . 3 (rec(𝑅, 𝐴) “ ω) = ran (rec(𝑅, 𝐴) ↾ ω)
252251unieqi 4381 . 2 (rec(𝑅, 𝐴) “ ω) = ran (rec(𝑅, 𝐴) ↾ ω)
253250, 252syl6eqr 2662 1 (𝐴𝑉 → (fi‘𝐴) = (rec(𝑅, 𝐴) “ ω))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372  ∩ cint 4410  ∪ ciun 4455   ↦ cmpt 4643   × cxp 5036  ran crn 5039   ↾ cres 5040   “ cima 5041  Ord word 5639  Oncon0 5640  suc csuc 5642   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804   ↦ cmpt2 6551  ωcom 6957  reccrdg 7392  ficfi 8199 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200 This theorem is referenced by: (None)
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