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Theorem dffv2 6181
 Description: Alternate definition of function value df-fv 5812 that doesn't require dummy variables. (Contributed by NM, 4-Aug-2010.)
Assertion
Ref Expression
dffv2 (𝐹𝐴) = ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))

Proof of Theorem dffv2
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snidb 4154 . . . . 5 (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
2 fvres 6117 . . . . 5 (𝐴 ∈ {𝐴} → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹𝐴))
31, 2sylbi 206 . . . 4 (𝐴 ∈ V → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹𝐴))
4 fvprc 6097 . . . . 5 𝐴 ∈ V → ((𝐹 ↾ {𝐴})‘𝐴) = ∅)
5 fvprc 6097 . . . . 5 𝐴 ∈ V → (𝐹𝐴) = ∅)
64, 5eqtr4d 2647 . . . 4 𝐴 ∈ V → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹𝐴))
73, 6pm2.61i 175 . . 3 ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹𝐴)
8 funfv 6175 . . . 4 (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴})‘𝐴) = ((𝐹 ↾ {𝐴}) “ {𝐴}))
9 df-fun 5806 . . . . . . . . . . . . 13 (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ⊆ I ))
109simprbi 479 . . . . . . . . . . . 12 (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ⊆ I )
11 ssdif0 3896 . . . . . . . . . . . 12 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ⊆ I ↔ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) = ∅)
1210, 11sylib 207 . . . . . . . . . . 11 (Fun (𝐹 ↾ {𝐴}) → (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) = ∅)
1312unieqd 4382 . . . . . . . . . 10 (Fun (𝐹 ↾ {𝐴}) → (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) = ∅)
14 uni0 4401 . . . . . . . . . 10 ∅ = ∅
1513, 14syl6eq 2660 . . . . . . . . 9 (Fun (𝐹 ↾ {𝐴}) → (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) = ∅)
1615unieqd 4382 . . . . . . . 8 (Fun (𝐹 ↾ {𝐴}) → (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) = ∅)
1716, 14syl6eq 2660 . . . . . . 7 (Fun (𝐹 ↾ {𝐴}) → (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) = ∅)
1817difeq2d 3690 . . . . . 6 (Fun (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ((𝐹 “ {𝐴}) ∖ ∅))
19 resima 5351 . . . . . . 7 ((𝐹 ↾ {𝐴}) “ {𝐴}) = (𝐹 “ {𝐴})
20 dif0 3904 . . . . . . 7 ((𝐹 “ {𝐴}) ∖ ∅) = (𝐹 “ {𝐴})
2119, 20eqtr4i 2635 . . . . . 6 ((𝐹 ↾ {𝐴}) “ {𝐴}) = ((𝐹 “ {𝐴}) ∖ ∅)
2218, 21syl6reqr 2663 . . . . 5 (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴}) “ {𝐴}) = ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
2322unieqd 4382 . . . 4 (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴}) “ {𝐴}) = ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
248, 23eqtrd 2644 . . 3 (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴})‘𝐴) = ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
257, 24syl5eqr 2658 . 2 (Fun (𝐹 ↾ {𝐴}) → (𝐹𝐴) = ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
26 nfunsn 6135 . . 3 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹𝐴) = ∅)
27 relres 5346 . . . . . . . . . . . . . . 15 Rel (𝐹 ↾ {𝐴})
28 dffun3 5815 . . . . . . . . . . . . . . 15 (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧𝑧 = 𝑦)))
2927, 28mpbiran 955 . . . . . . . . . . . . . 14 (Fun (𝐹 ↾ {𝐴}) ↔ ∀𝑥𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧𝑧 = 𝑦))
30 iman 439 . . . . . . . . . . . . . . . . . . 19 ((𝑥(𝐹 ↾ {𝐴})𝑧𝑧 = 𝑦) ↔ ¬ (𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
3130albii 1737 . . . . . . . . . . . . . . . . . 18 (∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧𝑧 = 𝑦) ↔ ∀𝑧 ¬ (𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
32 alnex 1697 . . . . . . . . . . . . . . . . . 18 (∀𝑧 ¬ (𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ↔ ¬ ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
3331, 32bitri 263 . . . . . . . . . . . . . . . . 17 (∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧𝑧 = 𝑦) ↔ ¬ ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
3433exbii 1764 . . . . . . . . . . . . . . . 16 (∃𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧𝑧 = 𝑦) ↔ ∃𝑦 ¬ ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
35 exnal 1744 . . . . . . . . . . . . . . . 16 (∃𝑦 ¬ ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ↔ ¬ ∀𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
3634, 35bitri 263 . . . . . . . . . . . . . . 15 (∃𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧𝑧 = 𝑦) ↔ ¬ ∀𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
3736albii 1737 . . . . . . . . . . . . . 14 (∀𝑥𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧𝑧 = 𝑦) ↔ ∀𝑥 ¬ ∀𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
38 alnex 1697 . . . . . . . . . . . . . 14 (∀𝑥 ¬ ∀𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ↔ ¬ ∃𝑥𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
3929, 37, 383bitrri 286 . . . . . . . . . . . . 13 (¬ ∃𝑥𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ↔ Fun (𝐹 ↾ {𝐴}))
4039con1bii 345 . . . . . . . . . . . 12 (¬ Fun (𝐹 ↾ {𝐴}) ↔ ∃𝑥𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
41 sp 2041 . . . . . . . . . . . . 13 (∀𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
4241eximi 1752 . . . . . . . . . . . 12 (∃𝑥𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → ∃𝑥𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
4340, 42sylbi 206 . . . . . . . . . . 11 (¬ Fun (𝐹 ↾ {𝐴}) → ∃𝑥𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
44 snssi 4280 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → {𝐴} ⊆ dom (𝐹 ↾ {𝐴}))
45 residm 5350 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = (𝐹 ↾ {𝐴})
4645dmeqi 5247 . . . . . . . . . . . . . . . . . . . . 21 dom ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = dom (𝐹 ↾ {𝐴})
47 ssdmres 5340 . . . . . . . . . . . . . . . . . . . . . 22 ({𝐴} ⊆ dom (𝐹 ↾ {𝐴}) ↔ dom ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = {𝐴})
4847biimpi 205 . . . . . . . . . . . . . . . . . . . . 21 ({𝐴} ⊆ dom (𝐹 ↾ {𝐴}) → dom ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = {𝐴})
4946, 48syl5eqr 2658 . . . . . . . . . . . . . . . . . . . 20 ({𝐴} ⊆ dom (𝐹 ↾ {𝐴}) → dom (𝐹 ↾ {𝐴}) = {𝐴})
5044, 49syl 17 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → dom (𝐹 ↾ {𝐴}) = {𝐴})
51 vex 3176 . . . . . . . . . . . . . . . . . . . 20 𝑥 ∈ V
52 vex 3176 . . . . . . . . . . . . . . . . . . . 20 𝑧 ∈ V
5351, 52breldm 5251 . . . . . . . . . . . . . . . . . . 19 (𝑥(𝐹 ↾ {𝐴})𝑧𝑥 ∈ dom (𝐹 ↾ {𝐴}))
54 eleq2 2677 . . . . . . . . . . . . . . . . . . . . 21 (dom (𝐹 ↾ {𝐴}) = {𝐴} → (𝑥 ∈ dom (𝐹 ↾ {𝐴}) ↔ 𝑥 ∈ {𝐴}))
55 velsn 4141 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
5654, 55syl6bb 275 . . . . . . . . . . . . . . . . . . . 20 (dom (𝐹 ↾ {𝐴}) = {𝐴} → (𝑥 ∈ dom (𝐹 ↾ {𝐴}) ↔ 𝑥 = 𝐴))
5756biimpa 500 . . . . . . . . . . . . . . . . . . 19 ((dom (𝐹 ↾ {𝐴}) = {𝐴} ∧ 𝑥 ∈ dom (𝐹 ↾ {𝐴})) → 𝑥 = 𝐴)
5850, 53, 57syl2an 493 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ dom (𝐹 ↾ {𝐴}) ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) → 𝑥 = 𝐴)
5958breq1d 4593 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ dom (𝐹 ↾ {𝐴}) ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) → (𝑥(𝐹 ↾ {𝐴})𝑧𝐴(𝐹 ↾ {𝐴})𝑧))
6059biimpd 218 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ dom (𝐹 ↾ {𝐴}) ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) → (𝑥(𝐹 ↾ {𝐴})𝑧𝐴(𝐹 ↾ {𝐴})𝑧))
6160ex 449 . . . . . . . . . . . . . . 15 (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (𝑥(𝐹 ↾ {𝐴})𝑧 → (𝑥(𝐹 ↾ {𝐴})𝑧𝐴(𝐹 ↾ {𝐴})𝑧)))
6261pm2.43d 51 . . . . . . . . . . . . . 14 (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (𝑥(𝐹 ↾ {𝐴})𝑧𝐴(𝐹 ↾ {𝐴})𝑧))
6362anim1d 586 . . . . . . . . . . . . 13 (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → (𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦)))
6463eximdv 1833 . . . . . . . . . . . 12 (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → ∃𝑧(𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦)))
6564exlimdv 1848 . . . . . . . . . . 11 (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (∃𝑥𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → ∃𝑧(𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦)))
6643, 65mpan9 485 . . . . . . . . . 10 ((¬ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → ∃𝑧(𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
6719eleq2i 2680 . . . . . . . . . . . . 13 (𝑦 ∈ ((𝐹 ↾ {𝐴}) “ {𝐴}) ↔ 𝑦 ∈ (𝐹 “ {𝐴}))
68 elimasni 5411 . . . . . . . . . . . . 13 (𝑦 ∈ ((𝐹 ↾ {𝐴}) “ {𝐴}) → 𝐴(𝐹 ↾ {𝐴})𝑦)
6967, 68sylbir 224 . . . . . . . . . . . 12 (𝑦 ∈ (𝐹 “ {𝐴}) → 𝐴(𝐹 ↾ {𝐴})𝑦)
70 vex 3176 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
7170, 52uniop 4902 . . . . . . . . . . . . . . . 16 𝑦, 𝑧⟩ = {𝑦, 𝑧}
72 opex 4859 . . . . . . . . . . . . . . . . . . 19 𝑦, 𝑧⟩ ∈ V
7372unisn 4387 . . . . . . . . . . . . . . . . . 18 {⟨𝑦, 𝑧⟩} = ⟨𝑦, 𝑧
7427brrelexi 5082 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴(𝐹 ↾ {𝐴})𝑧𝐴 ∈ V)
75 brcnvg 5225 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∈ V ∧ 𝐴 ∈ V) → (𝑦(𝐹 ↾ {𝐴})𝐴𝐴(𝐹 ↾ {𝐴})𝑦))
7670, 74, 75sylancr 694 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴(𝐹 ↾ {𝐴})𝑧 → (𝑦(𝐹 ↾ {𝐴})𝐴𝐴(𝐹 ↾ {𝐴})𝑦))
7776biimpar 501 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴(𝐹 ↾ {𝐴})𝑧𝐴(𝐹 ↾ {𝐴})𝑦) → 𝑦(𝐹 ↾ {𝐴})𝐴)
7874adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦(𝐹 ↾ {𝐴})𝐴𝐴(𝐹 ↾ {𝐴})𝑧) → 𝐴 ∈ V)
79 breq2 4587 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = 𝐴 → (𝑦(𝐹 ↾ {𝐴})𝑥𝑦(𝐹 ↾ {𝐴})𝐴))
80 breq1 4586 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = 𝐴 → (𝑥(𝐹 ↾ {𝐴})𝑧𝐴(𝐹 ↾ {𝐴})𝑧))
8179, 80anbi12d 743 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝐴 → ((𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧) ↔ (𝑦(𝐹 ↾ {𝐴})𝐴𝐴(𝐹 ↾ {𝐴})𝑧)))
8281rspcev 3282 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ V ∧ (𝑦(𝐹 ↾ {𝐴})𝐴𝐴(𝐹 ↾ {𝐴})𝑧)) → ∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧))
8378, 82mpancom 700 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦(𝐹 ↾ {𝐴})𝐴𝐴(𝐹 ↾ {𝐴})𝑧) → ∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧))
8483ancoms 468 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴(𝐹 ↾ {𝐴})𝑧𝑦(𝐹 ↾ {𝐴})𝐴) → ∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧))
8577, 84syldan 486 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴(𝐹 ↾ {𝐴})𝑧𝐴(𝐹 ↾ {𝐴})𝑦) → ∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧))
8685anim1i 590 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴(𝐹 ↾ {𝐴})𝑧𝐴(𝐹 ↾ {𝐴})𝑦) ∧ ¬ 𝑧 = 𝑦) → (∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧) ∧ ¬ 𝑧 = 𝑦))
8786an32s 842 . . . . . . . . . . . . . . . . . . . 20 (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → (∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧) ∧ ¬ 𝑧 = 𝑦))
88 eldif 3550 . . . . . . . . . . . . . . . . . . . . 21 (⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) ↔ (⟨𝑦, 𝑧⟩ ∈ ((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∧ ¬ ⟨𝑦, 𝑧⟩ ∈ I ))
89 rexv 3193 . . . . . . . . . . . . . . . . . . . . . . 23 (∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧) ↔ ∃𝑥(𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧))
9070, 52brco 5214 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴}))𝑧 ↔ ∃𝑥(𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧))
91 df-br 4584 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴}))𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})))
9289, 90, 913bitr2ri 288 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝑦, 𝑧⟩ ∈ ((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ↔ ∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧))
9352ideq 5196 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 I 𝑧𝑦 = 𝑧)
94 df-br 4584 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 I 𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ I )
95 equcom 1932 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑧𝑧 = 𝑦)
9693, 94, 953bitr3i 289 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝑦, 𝑧⟩ ∈ I ↔ 𝑧 = 𝑦)
9796notbii 309 . . . . . . . . . . . . . . . . . . . . . 22 (¬ ⟨𝑦, 𝑧⟩ ∈ I ↔ ¬ 𝑧 = 𝑦)
9892, 97anbi12i 729 . . . . . . . . . . . . . . . . . . . . 21 ((⟨𝑦, 𝑧⟩ ∈ ((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∧ ¬ ⟨𝑦, 𝑧⟩ ∈ I ) ↔ (∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧) ∧ ¬ 𝑧 = 𝑦))
9988, 98bitr2i 264 . . . . . . . . . . . . . . . . . . . 20 ((∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧) ∧ ¬ 𝑧 = 𝑦) ↔ ⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
10087, 99sylib 207 . . . . . . . . . . . . . . . . . . 19 (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → ⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
101 snssi 4280 . . . . . . . . . . . . . . . . . . 19 (⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) → {⟨𝑦, 𝑧⟩} ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
102 uniss 4394 . . . . . . . . . . . . . . . . . . 19 ({⟨𝑦, 𝑧⟩} ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) → {⟨𝑦, 𝑧⟩} ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
103100, 101, 1023syl 18 . . . . . . . . . . . . . . . . . 18 (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → {⟨𝑦, 𝑧⟩} ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
10473, 103syl5eqssr 3613 . . . . . . . . . . . . . . . . 17 (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → ⟨𝑦, 𝑧⟩ ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
105104unissd 4398 . . . . . . . . . . . . . . . 16 (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → 𝑦, 𝑧⟩ ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
10671, 105syl5eqssr 3613 . . . . . . . . . . . . . . 15 (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → {𝑦, 𝑧} ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
10770, 52prss 4291 . . . . . . . . . . . . . . 15 ((𝑦 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) ∧ 𝑧 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) ↔ {𝑦, 𝑧} ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
108106, 107sylibr 223 . . . . . . . . . . . . . 14 (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → (𝑦 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) ∧ 𝑧 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
109108simpld 474 . . . . . . . . . . . . 13 (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → 𝑦 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
110109ex 449 . . . . . . . . . . . 12 ((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → (𝐴(𝐹 ↾ {𝐴})𝑦𝑦 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
11169, 110syl5 33 . . . . . . . . . . 11 ((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → (𝑦 ∈ (𝐹 “ {𝐴}) → 𝑦 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
112111exlimiv 1845 . . . . . . . . . 10 (∃𝑧(𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → (𝑦 ∈ (𝐹 “ {𝐴}) → 𝑦 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
11366, 112syl 17 . . . . . . . . 9 ((¬ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → (𝑦 ∈ (𝐹 “ {𝐴}) → 𝑦 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
114113ssrdv 3574 . . . . . . . 8 ((¬ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → (𝐹 “ {𝐴}) ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
115 ssdif0 3896 . . . . . . . 8 ((𝐹 “ {𝐴}) ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) ↔ ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ∅)
116114, 115sylib 207 . . . . . . 7 ((¬ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ∅)
117116ex 449 . . . . . 6 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ∅))
118 ndmima 5421 . . . . . . . . 9 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴}) “ {𝐴}) = ∅)
11919, 118syl5eqr 2658 . . . . . . . 8 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (𝐹 “ {𝐴}) = ∅)
120119difeq1d 3689 . . . . . . 7 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = (∅ ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
121 0dif 3929 . . . . . . 7 (∅ ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ∅
122120, 121syl6eq 2660 . . . . . 6 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ∅)
123117, 122pm2.61d1 170 . . . . 5 (¬ Fun (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ∅)
124123unieqd 4382 . . . 4 (¬ Fun (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ∅)
125124, 14syl6eq 2660 . . 3 (¬ Fun (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ∅)
12626, 125eqtr4d 2647 . 2 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹𝐴) = ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
12725, 126pm2.61i 175 1 (𝐹𝐴) = ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  {csn 4125  {cpr 4127  ⟨cop 4131  ∪ cuni 4372   class class class wbr 4583   I cid 4948  ◡ccnv 5037  dom cdm 5038   ↾ cres 5040   “ cima 5041   ∘ ccom 5042  Rel wrel 5043  Fun wfun 5798  ‘cfv 5804 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 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  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-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812 This theorem is referenced by: (None)
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