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Theorem fpwwe2 9344
 Description: Given any function 𝐹 from well-orderings of subsets of 𝐴 to 𝐴, there is a unique well-ordered subset ⟨𝑋, (𝑊‘𝑋)⟩ which "agrees" with 𝐹 in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 8736. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.)
Hypotheses
Ref Expression
fpwwe2.1 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
fpwwe2.2 (𝜑𝐴 ∈ V)
fpwwe2.3 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
fpwwe2.4 𝑋 = dom 𝑊
Assertion
Ref Expression
fpwwe2 (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
Distinct variable groups:   𝑦,𝑢,𝑟,𝑥,𝐹   𝑋,𝑟,𝑢,𝑥,𝑦   𝜑,𝑟,𝑢,𝑥,𝑦   𝐴,𝑟,𝑥   𝑅,𝑟,𝑢,𝑥,𝑦   𝑌,𝑟,𝑢,𝑥,𝑦   𝑊,𝑟,𝑢,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑦,𝑢)

Proof of Theorem fpwwe2
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fpwwe2.1 . . . . . . . . . . 11 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
2 fpwwe2.2 . . . . . . . . . . 11 (𝜑𝐴 ∈ V)
3 fpwwe2.3 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
4 fpwwe2.4 . . . . . . . . . . 11 𝑋 = dom 𝑊
51, 2, 3, 4fpwwe2lem11 9341 . . . . . . . . . 10 (𝜑𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋))
6 ffun 5961 . . . . . . . . . 10 (𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋) → Fun 𝑊)
75, 6syl 17 . . . . . . . . 9 (𝜑 → Fun 𝑊)
8 funbrfv2b 6150 . . . . . . . . 9 (Fun 𝑊 → (𝑌𝑊𝑅 ↔ (𝑌 ∈ dom 𝑊 ∧ (𝑊𝑌) = 𝑅)))
97, 8syl 17 . . . . . . . 8 (𝜑 → (𝑌𝑊𝑅 ↔ (𝑌 ∈ dom 𝑊 ∧ (𝑊𝑌) = 𝑅)))
109simprbda 651 . . . . . . 7 ((𝜑𝑌𝑊𝑅) → 𝑌 ∈ dom 𝑊)
1110adantrr 749 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌 ∈ dom 𝑊)
12 elssuni 4403 . . . . . . 7 (𝑌 ∈ dom 𝑊𝑌 dom 𝑊)
1312, 4syl6sseqr 3615 . . . . . 6 (𝑌 ∈ dom 𝑊𝑌𝑋)
1411, 13syl 17 . . . . 5 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌𝑋)
15 simpl 472 . . . . . . 7 ((𝑋𝑌 ∧ (𝑊𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) → 𝑋𝑌)
1615a1i 11 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑋𝑌 ∧ (𝑊𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) → 𝑋𝑌))
17 simplrr 797 . . . . . . . . 9 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑌𝐹𝑅) ∈ 𝑌)
182adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝐴 ∈ V)
1918adantr 480 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝐴 ∈ V)
201, 2, 3, 4fpwwe2lem12 9342 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑋 ∈ dom 𝑊)
21 funfvbrb 6238 . . . . . . . . . . . . . . . . . . . 20 (Fun 𝑊 → (𝑋 ∈ dom 𝑊𝑋𝑊(𝑊𝑋)))
227, 21syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑋 ∈ dom 𝑊𝑋𝑊(𝑊𝑋)))
2320, 22mpbid 221 . . . . . . . . . . . . . . . . . 18 (𝜑𝑋𝑊(𝑊𝑋))
241, 2fpwwe2lem2 9333 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑋𝑊(𝑊𝑋) ↔ ((𝑋𝐴 ∧ (𝑊𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊𝑋) We 𝑋 ∧ ∀𝑦𝑋 [((𝑊𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝑋) ∩ (𝑢 × 𝑢))) = 𝑦))))
2523, 24mpbid 221 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑋𝐴 ∧ (𝑊𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊𝑋) We 𝑋 ∧ ∀𝑦𝑋 [((𝑊𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝑋) ∩ (𝑢 × 𝑢))) = 𝑦)))
2625ad2antrr 758 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋𝐴 ∧ (𝑊𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊𝑋) We 𝑋 ∧ ∀𝑦𝑋 [((𝑊𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝑋) ∩ (𝑢 × 𝑢))) = 𝑦)))
2726simpld 474 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋𝐴 ∧ (𝑊𝑋) ⊆ (𝑋 × 𝑋)))
2827simpld 474 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋𝐴)
2919, 28ssexd 4733 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋 ∈ V)
30 difexg 4735 . . . . . . . . . . . . 13 (𝑋 ∈ V → (𝑋𝑌) ∈ V)
3129, 30syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋𝑌) ∈ V)
3226simprd 478 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑊𝑋) We 𝑋 ∧ ∀𝑦𝑋 [((𝑊𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝑋) ∩ (𝑢 × 𝑢))) = 𝑦))
3332simpld 474 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊𝑋) We 𝑋)
34 wefr 5028 . . . . . . . . . . . . 13 ((𝑊𝑋) We 𝑋 → (𝑊𝑋) Fr 𝑋)
3533, 34syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊𝑋) Fr 𝑋)
36 difssd 3700 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋𝑌) ⊆ 𝑋)
37 fri 5000 . . . . . . . . . . . . 13 ((((𝑋𝑌) ∈ V ∧ (𝑊𝑋) Fr 𝑋) ∧ ((𝑋𝑌) ⊆ 𝑋 ∧ (𝑋𝑌) ≠ ∅)) → ∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧)
3837expr 641 . . . . . . . . . . . 12 ((((𝑋𝑌) ∈ V ∧ (𝑊𝑋) Fr 𝑋) ∧ (𝑋𝑌) ⊆ 𝑋) → ((𝑋𝑌) ≠ ∅ → ∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧))
3931, 35, 36, 38syl21anc 1317 . . . . . . . . . . 11 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋𝑌) ≠ ∅ → ∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧))
40 ssdif0 3896 . . . . . . . . . . . . . . 15 ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ⊆ 𝑌 ↔ ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ∖ 𝑌) = ∅)
41 indif1 3830 . . . . . . . . . . . . . . . 16 ((𝑋𝑌) ∩ ((𝑊𝑋) “ {𝑧})) = ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ∖ 𝑌)
4241eqeq1i 2615 . . . . . . . . . . . . . . 15 (((𝑋𝑌) ∩ ((𝑊𝑋) “ {𝑧})) = ∅ ↔ ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ∖ 𝑌) = ∅)
43 disj 3969 . . . . . . . . . . . . . . . 16 (((𝑋𝑌) ∩ ((𝑊𝑋) “ {𝑧})) = ∅ ↔ ∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤 ∈ ((𝑊𝑋) “ {𝑧}))
44 vex 3176 . . . . . . . . . . . . . . . . . . 19 𝑧 ∈ V
45 vex 3176 . . . . . . . . . . . . . . . . . . . 20 𝑤 ∈ V
4645eliniseg 5413 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ V → (𝑤 ∈ ((𝑊𝑋) “ {𝑧}) ↔ 𝑤(𝑊𝑋)𝑧))
4744, 46ax-mp 5 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ((𝑊𝑋) “ {𝑧}) ↔ 𝑤(𝑊𝑋)𝑧)
4847notbii 309 . . . . . . . . . . . . . . . . 17 𝑤 ∈ ((𝑊𝑋) “ {𝑧}) ↔ ¬ 𝑤(𝑊𝑋)𝑧)
4948ralbii 2963 . . . . . . . . . . . . . . . 16 (∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤 ∈ ((𝑊𝑋) “ {𝑧}) ↔ ∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧)
5043, 49bitri 263 . . . . . . . . . . . . . . 15 (((𝑋𝑌) ∩ ((𝑊𝑋) “ {𝑧})) = ∅ ↔ ∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧)
5140, 42, 503bitr2i 287 . . . . . . . . . . . . . 14 ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ⊆ 𝑌 ↔ ∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧)
52 cnvimass 5404 . . . . . . . . . . . . . . . . 17 ((𝑊𝑋) “ {𝑧}) ⊆ dom (𝑊𝑋)
5327simprd 478 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊𝑋) ⊆ (𝑋 × 𝑋))
54 dmss 5245 . . . . . . . . . . . . . . . . . . 19 ((𝑊𝑋) ⊆ (𝑋 × 𝑋) → dom (𝑊𝑋) ⊆ dom (𝑋 × 𝑋))
5553, 54syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → dom (𝑊𝑋) ⊆ dom (𝑋 × 𝑋))
56 dmxpid 5266 . . . . . . . . . . . . . . . . . 18 dom (𝑋 × 𝑋) = 𝑋
5755, 56syl6sseq 3614 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → dom (𝑊𝑋) ⊆ 𝑋)
5852, 57syl5ss 3579 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑊𝑋) “ {𝑧}) ⊆ 𝑋)
59 sseqin2 3779 . . . . . . . . . . . . . . . 16 (((𝑊𝑋) “ {𝑧}) ⊆ 𝑋 ↔ (𝑋 ∩ ((𝑊𝑋) “ {𝑧})) = ((𝑊𝑋) “ {𝑧}))
6058, 59sylib 207 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋 ∩ ((𝑊𝑋) “ {𝑧})) = ((𝑊𝑋) “ {𝑧}))
6160sseq1d 3595 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ⊆ 𝑌 ↔ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌))
6251, 61syl5bbr 273 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧 ↔ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌))
6362rexbidv 3034 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧 ↔ ∃𝑧 ∈ (𝑋𝑌)((𝑊𝑋) “ {𝑧}) ⊆ 𝑌))
64 eldifn 3695 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 ∈ (𝑋𝑌) → ¬ 𝑧𝑌)
6564ad2antrl 760 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ¬ 𝑧𝑌)
66 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = 𝑧 → (𝑤𝑌𝑧𝑌))
6766notbid 307 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 𝑧 → (¬ 𝑤𝑌 ↔ ¬ 𝑧𝑌))
6865, 67syl5ibrcom 236 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑤 = 𝑧 → ¬ 𝑤𝑌))
6968con2d 128 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑤𝑌 → ¬ 𝑤 = 𝑧))
7069imp 444 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → ¬ 𝑤 = 𝑧)
7165adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → ¬ 𝑧𝑌)
72 simprr 792 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))
7372ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))
7473breqd 4594 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧𝑅𝑤𝑧((𝑊𝑋) ∩ (𝑋 × 𝑌))𝑤))
75 eldifi 3694 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ (𝑋𝑌) → 𝑧𝑋)
7675ad2antrl 760 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑧𝑋)
7776adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑧𝑋)
78 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑤𝑌)
79 brxp 5071 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧(𝑋 × 𝑌)𝑤 ↔ (𝑧𝑋𝑤𝑌))
8077, 78, 79sylanbrc 695 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑧(𝑋 × 𝑌)𝑤)
81 brin 4634 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧((𝑊𝑋) ∩ (𝑋 × 𝑌))𝑤 ↔ (𝑧(𝑊𝑋)𝑤𝑧(𝑋 × 𝑌)𝑤))
8281rbaib 945 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧(𝑋 × 𝑌)𝑤 → (𝑧((𝑊𝑋) ∩ (𝑋 × 𝑌))𝑤𝑧(𝑊𝑋)𝑤))
8380, 82syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧((𝑊𝑋) ∩ (𝑋 × 𝑌))𝑤𝑧(𝑊𝑋)𝑤))
8474, 83bitrd 267 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧𝑅𝑤𝑧(𝑊𝑋)𝑤))
851, 2fpwwe2lem2 9333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → (𝑌𝑊𝑅 ↔ ((𝑌𝐴𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦𝑌 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
8685biimpa 500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑌𝑊𝑅) → ((𝑌𝐴𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦𝑌 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
8786adantrr 749 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑌𝐴𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦𝑌 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
8887simpld 474 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑌𝐴𝑅 ⊆ (𝑌 × 𝑌)))
8988simprd 478 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑅 ⊆ (𝑌 × 𝑌))
9089ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑅 ⊆ (𝑌 × 𝑌))
9190ssbrd 4626 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧𝑅𝑤𝑧(𝑌 × 𝑌)𝑤))
92 brxp 5071 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧(𝑌 × 𝑌)𝑤 ↔ (𝑧𝑌𝑤𝑌))
9392simplbi 475 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧(𝑌 × 𝑌)𝑤𝑧𝑌)
9491, 93syl6 34 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧𝑅𝑤𝑧𝑌))
9584, 94sylbird 249 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧(𝑊𝑋)𝑤𝑧𝑌))
9671, 95mtod 188 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → ¬ 𝑧(𝑊𝑋)𝑤)
9733ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑊𝑋) We 𝑋)
98 weso 5029 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑊𝑋) We 𝑋 → (𝑊𝑋) Or 𝑋)
9997, 98syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑊𝑋) Or 𝑋)
10014ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌𝑋)
101100sselda 3568 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑤𝑋)
102 sotric 4985 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑊𝑋) Or 𝑋 ∧ (𝑤𝑋𝑧𝑋)) → (𝑤(𝑊𝑋)𝑧 ↔ ¬ (𝑤 = 𝑧𝑧(𝑊𝑋)𝑤)))
103 ioran 510 . . . . . . . . . . . . . . . . . . . . . . 23 (¬ (𝑤 = 𝑧𝑧(𝑊𝑋)𝑤) ↔ (¬ 𝑤 = 𝑧 ∧ ¬ 𝑧(𝑊𝑋)𝑤))
104102, 103syl6bb 275 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑊𝑋) Or 𝑋 ∧ (𝑤𝑋𝑧𝑋)) → (𝑤(𝑊𝑋)𝑧 ↔ (¬ 𝑤 = 𝑧 ∧ ¬ 𝑧(𝑊𝑋)𝑤)))
10599, 101, 77, 104syl12anc 1316 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑤(𝑊𝑋)𝑧 ↔ (¬ 𝑤 = 𝑧 ∧ ¬ 𝑧(𝑊𝑋)𝑤)))
10670, 96, 105mpbir2and 959 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑤(𝑊𝑋)𝑧)
107106, 47sylibr 223 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑤 ∈ ((𝑊𝑋) “ {𝑧}))
108107ex 449 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑤𝑌𝑤 ∈ ((𝑊𝑋) “ {𝑧})))
109108ssrdv 3574 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌 ⊆ ((𝑊𝑋) “ {𝑧}))
110 simprr 792 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)
111109, 110eqssd 3585 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌 = ((𝑊𝑋) “ {𝑧}))
112 in32 3787 . . . . . . . . . . . . . . . . . 18 (((𝑊𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)) = (((𝑊𝑋) ∩ (𝑌 × 𝑌)) ∩ (𝑋 × 𝑌))
113 simplrr 797 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))
114113ineq1d 3775 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑅 ∩ (𝑌 × 𝑌)) = (((𝑊𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)))
11589ad2antrr 758 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 ⊆ (𝑌 × 𝑌))
116 df-ss 3554 . . . . . . . . . . . . . . . . . . . 20 (𝑅 ⊆ (𝑌 × 𝑌) ↔ (𝑅 ∩ (𝑌 × 𝑌)) = 𝑅)
117115, 116sylib 207 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑅 ∩ (𝑌 × 𝑌)) = 𝑅)
118114, 117eqtr3d 2646 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (((𝑊𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)) = 𝑅)
119 inss2 3796 . . . . . . . . . . . . . . . . . . . 20 ((𝑊𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑌 × 𝑌)
120 xpss1 5151 . . . . . . . . . . . . . . . . . . . . 21 (𝑌𝑋 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑌))
121100, 120syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑌))
122119, 121syl5ss 3579 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ((𝑊𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑋 × 𝑌))
123 df-ss 3554 . . . . . . . . . . . . . . . . . . 19 (((𝑊𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑋 × 𝑌) ↔ (((𝑊𝑋) ∩ (𝑌 × 𝑌)) ∩ (𝑋 × 𝑌)) = ((𝑊𝑋) ∩ (𝑌 × 𝑌)))
124122, 123sylib 207 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (((𝑊𝑋) ∩ (𝑌 × 𝑌)) ∩ (𝑋 × 𝑌)) = ((𝑊𝑋) ∩ (𝑌 × 𝑌)))
125112, 118, 1243eqtr3a 2668 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊𝑋) ∩ (𝑌 × 𝑌)))
126111sqxpeqd 5065 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌 × 𝑌) = (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧})))
127126ineq2d 3776 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ((𝑊𝑋) ∩ (𝑌 × 𝑌)) = ((𝑊𝑋) ∩ (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧}))))
128125, 127eqtrd 2644 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊𝑋) ∩ (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧}))))
129111, 128oveq12d 6567 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌𝐹𝑅) = (((𝑊𝑋) “ {𝑧})𝐹((𝑊𝑋) ∩ (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧})))))
13019adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝐴 ∈ V)
13123adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑊(𝑊𝑋))
132131ad2antrr 758 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑋𝑊(𝑊𝑋))
1331, 130, 132fpwwe2lem3 9334 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑧𝑋) → (((𝑊𝑋) “ {𝑧})𝐹((𝑊𝑋) ∩ (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧})))) = 𝑧)
13476, 133mpdan 699 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (((𝑊𝑋) “ {𝑧})𝐹((𝑊𝑋) ∩ (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧})))) = 𝑧)
135129, 134eqtrd 2644 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌𝐹𝑅) = 𝑧)
136135, 65eqneltrd 2707 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ¬ (𝑌𝐹𝑅) ∈ 𝑌)
137136rexlimdvaa 3014 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (∃𝑧 ∈ (𝑋𝑌)((𝑊𝑋) “ {𝑧}) ⊆ 𝑌 → ¬ (𝑌𝐹𝑅) ∈ 𝑌))
13863, 137sylbid 229 . . . . . . . . . . 11 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧 → ¬ (𝑌𝐹𝑅) ∈ 𝑌))
13939, 138syld 46 . . . . . . . . . 10 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋𝑌) ≠ ∅ → ¬ (𝑌𝐹𝑅) ∈ 𝑌))
140139necon4ad 2801 . . . . . . . . 9 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑌𝐹𝑅) ∈ 𝑌 → (𝑋𝑌) = ∅))
14117, 140mpd 15 . . . . . . . 8 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋𝑌) = ∅)
142 ssdif0 3896 . . . . . . . 8 (𝑋𝑌 ↔ (𝑋𝑌) = ∅)
143141, 142sylibr 223 . . . . . . 7 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋𝑌)
144143ex 449 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌))) → 𝑋𝑌))
1453adantlr 747 . . . . . . 7 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
146 simprl 790 . . . . . . 7 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌𝑊𝑅)
1471, 18, 145, 131, 146fpwwe2lem10 9340 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑋𝑌 ∧ (𝑊𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) ∨ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))))
14816, 144, 147mpjaod 395 . . . . 5 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑌)
14914, 148eqssd 3585 . . . 4 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌 = 𝑋)
1507adantr 480 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → Fun 𝑊)
151149, 146eqbrtrrd 4607 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑊𝑅)
152 funbrfv 6144 . . . . . 6 (Fun 𝑊 → (𝑋𝑊𝑅 → (𝑊𝑋) = 𝑅))
153150, 151, 152sylc 63 . . . . 5 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑊𝑋) = 𝑅)
154153eqcomd 2616 . . . 4 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑅 = (𝑊𝑋))
155149, 154jca 553 . . 3 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑌 = 𝑋𝑅 = (𝑊𝑋)))
156155ex 449 . 2 (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) → (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
1571, 2, 3, 4fpwwe2lem13 9343 . . . 4 (𝜑 → (𝑋𝐹(𝑊𝑋)) ∈ 𝑋)
15823, 157jca 553 . . 3 (𝜑 → (𝑋𝑊(𝑊𝑋) ∧ (𝑋𝐹(𝑊𝑋)) ∈ 𝑋))
159 breq12 4588 . . . 4 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → (𝑌𝑊𝑅𝑋𝑊(𝑊𝑋)))
160 oveq12 6558 . . . . 5 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → (𝑌𝐹𝑅) = (𝑋𝐹(𝑊𝑋)))
161 simpl 472 . . . . 5 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → 𝑌 = 𝑋)
162160, 161eleq12d 2682 . . . 4 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → ((𝑌𝐹𝑅) ∈ 𝑌 ↔ (𝑋𝐹(𝑊𝑋)) ∈ 𝑋))
163159, 162anbi12d 743 . . 3 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑋𝑊(𝑊𝑋) ∧ (𝑋𝐹(𝑊𝑋)) ∈ 𝑋)))
164158, 163syl5ibrcom 236 . 2 (𝜑 → ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)))
165156, 164impbid 201 1 (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173  [wsbc 3402   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372   class class class wbr 4583  {copab 4642   Or wor 4958   Fr wfr 4994   We wwe 4996   × cxp 5036  ◡ccnv 5037  dom cdm 5038   “ cima 5041  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549 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-rep 4699  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-rmo 2904  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-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-se 4998  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-isom 5813  df-riota 6511  df-ov 6552  df-wrecs 7294  df-recs 7355  df-oi 8298 This theorem is referenced by:  fpwwe  9347  canthwelem  9351  pwfseqlem4  9363
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