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Theorem dffv2 6464
Description: Alternate definition of function value df-fv 6078 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 4367 . . . . 5 (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
2 fvres 6398 . . . . 5 (𝐴 ∈ {𝐴} → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹𝐴))
31, 2sylbi 208 . . . 4 (𝐴 ∈ V → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹𝐴))
4 fvprc 6372 . . . . 5 𝐴 ∈ V → ((𝐹 ↾ {𝐴})‘𝐴) = ∅)
5 fvprc 6372 . . . . 5 𝐴 ∈ V → (𝐹𝐴) = ∅)
64, 5eqtr4d 2802 . . . 4 𝐴 ∈ V → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹𝐴))
73, 6pm2.61i 176 . . 3 ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹𝐴)
8 funfv 6458 . . . 4 (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴})‘𝐴) = ((𝐹 ↾ {𝐴}) “ {𝐴}))
9 df-fun 6072 . . . . . . . . . . . . 13 (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ⊆ I ))
109simprbi 490 . . . . . . . . . . . 12 (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ⊆ I )
11 ssdif0 4108 . . . . . . . . . . . 12 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ⊆ I ↔ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) = ∅)
1210, 11sylib 209 . . . . . . . . . . 11 (Fun (𝐹 ↾ {𝐴}) → (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) = ∅)
1312unieqd 4606 . . . . . . . . . 10 (Fun (𝐹 ↾ {𝐴}) → (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) = ∅)
14 uni0 4625 . . . . . . . . . 10 ∅ = ∅
1513, 14syl6eq 2815 . . . . . . . . 9 (Fun (𝐹 ↾ {𝐴}) → (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) = ∅)
1615unieqd 4606 . . . . . . . 8 (Fun (𝐹 ↾ {𝐴}) → (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) = ∅)
1716, 14syl6eq 2815 . . . . . . 7 (Fun (𝐹 ↾ {𝐴}) → (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) = ∅)
1817difeq2d 3892 . . . . . 6 (Fun (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ((𝐹 “ {𝐴}) ∖ ∅))
19 resima 5608 . . . . . . 7 ((𝐹 ↾ {𝐴}) “ {𝐴}) = (𝐹 “ {𝐴})
20 dif0 4117 . . . . . . 7 ((𝐹 “ {𝐴}) ∖ ∅) = (𝐹 “ {𝐴})
2119, 20eqtr4i 2790 . . . . . 6 ((𝐹 ↾ {𝐴}) “ {𝐴}) = ((𝐹 “ {𝐴}) ∖ ∅)
2218, 21syl6reqr 2818 . . . . 5 (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴}) “ {𝐴}) = ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
2322unieqd 4606 . . . 4 (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴}) “ {𝐴}) = ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
248, 23eqtrd 2799 . . 3 (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴})‘𝐴) = ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
257, 24syl5eqr 2813 . 2 (Fun (𝐹 ↾ {𝐴}) → (𝐹𝐴) = ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
26 nfunsn 6417 . . 3 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹𝐴) = ∅)
27 relres 5603 . . . . . . . . . . . . . . 15 Rel (𝐹 ↾ {𝐴})
28 dffun3 6081 . . . . . . . . . . . . . . 15 (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧𝑧 = 𝑦)))
2927, 28mpbiran 700 . . . . . . . . . . . . . 14 (Fun (𝐹 ↾ {𝐴}) ↔ ∀𝑥𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧𝑧 = 𝑦))
30 iman 390 . . . . . . . . . . . . . . . . . . 19 ((𝑥(𝐹 ↾ {𝐴})𝑧𝑧 = 𝑦) ↔ ¬ (𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
3130albii 1914 . . . . . . . . . . . . . . . . . 18 (∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧𝑧 = 𝑦) ↔ ∀𝑧 ¬ (𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
32 alnex 1876 . . . . . . . . . . . . . . . . . 18 (∀𝑧 ¬ (𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ↔ ¬ ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
3331, 32bitri 266 . . . . . . . . . . . . . . . . 17 (∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧𝑧 = 𝑦) ↔ ¬ ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
3433exbii 1943 . . . . . . . . . . . . . . . 16 (∃𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧𝑧 = 𝑦) ↔ ∃𝑦 ¬ ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
35 exnal 1921 . . . . . . . . . . . . . . . 16 (∃𝑦 ¬ ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ↔ ¬ ∀𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
3634, 35bitri 266 . . . . . . . . . . . . . . 15 (∃𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧𝑧 = 𝑦) ↔ ¬ ∀𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
3736albii 1914 . . . . . . . . . . . . . 14 (∀𝑥𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧𝑧 = 𝑦) ↔ ∀𝑥 ¬ ∀𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
38 alnex 1876 . . . . . . . . . . . . . 14 (∀𝑥 ¬ ∀𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ↔ ¬ ∃𝑥𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
3929, 37, 383bitrri 289 . . . . . . . . . . . . 13 (¬ ∃𝑥𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ↔ Fun (𝐹 ↾ {𝐴}))
4039con1bii 347 . . . . . . . . . . . 12 (¬ Fun (𝐹 ↾ {𝐴}) ↔ ∃𝑥𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
41 sp 2215 . . . . . . . . . . . . 13 (∀𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
4241eximi 1929 . . . . . . . . . . . 12 (∃𝑥𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → ∃𝑥𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
4340, 42sylbi 208 . . . . . . . . . . 11 (¬ Fun (𝐹 ↾ {𝐴}) → ∃𝑥𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
44 snssi 4495 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → {𝐴} ⊆ dom (𝐹 ↾ {𝐴}))
45 residm 5607 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = (𝐹 ↾ {𝐴})
4645dmeqi 5495 . . . . . . . . . . . . . . . . . . . . 21 dom ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = dom (𝐹 ↾ {𝐴})
47 ssdmres 5597 . . . . . . . . . . . . . . . . . . . . . 22 ({𝐴} ⊆ dom (𝐹 ↾ {𝐴}) ↔ dom ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = {𝐴})
4847biimpi 207 . . . . . . . . . . . . . . . . . . . . 21 ({𝐴} ⊆ dom (𝐹 ↾ {𝐴}) → dom ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = {𝐴})
4946, 48syl5eqr 2813 . . . . . . . . . . . . . . . . . . . 20 ({𝐴} ⊆ dom (𝐹 ↾ {𝐴}) → dom (𝐹 ↾ {𝐴}) = {𝐴})
5044, 49syl 17 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → dom (𝐹 ↾ {𝐴}) = {𝐴})
51 vex 3353 . . . . . . . . . . . . . . . . . . . 20 𝑥 ∈ V
52 vex 3353 . . . . . . . . . . . . . . . . . . . 20 𝑧 ∈ V
5351, 52breldm 5499 . . . . . . . . . . . . . . . . . . 19 (𝑥(𝐹 ↾ {𝐴})𝑧𝑥 ∈ dom (𝐹 ↾ {𝐴}))
54 eleq2 2833 . . . . . . . . . . . . . . . . . . . . 21 (dom (𝐹 ↾ {𝐴}) = {𝐴} → (𝑥 ∈ dom (𝐹 ↾ {𝐴}) ↔ 𝑥 ∈ {𝐴}))
55 velsn 4352 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
5654, 55syl6bb 278 . . . . . . . . . . . . . . . . . . . 20 (dom (𝐹 ↾ {𝐴}) = {𝐴} → (𝑥 ∈ dom (𝐹 ↾ {𝐴}) ↔ 𝑥 = 𝐴))
5756biimpa 468 . . . . . . . . . . . . . . . . . . 19 ((dom (𝐹 ↾ {𝐴}) = {𝐴} ∧ 𝑥 ∈ dom (𝐹 ↾ {𝐴})) → 𝑥 = 𝐴)
5850, 53, 57syl2an 589 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ dom (𝐹 ↾ {𝐴}) ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) → 𝑥 = 𝐴)
5958breq1d 4821 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ dom (𝐹 ↾ {𝐴}) ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) → (𝑥(𝐹 ↾ {𝐴})𝑧𝐴(𝐹 ↾ {𝐴})𝑧))
6059biimpd 220 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ dom (𝐹 ↾ {𝐴}) ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) → (𝑥(𝐹 ↾ {𝐴})𝑧𝐴(𝐹 ↾ {𝐴})𝑧))
6160ex 401 . . . . . . . . . . . . . . 15 (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (𝑥(𝐹 ↾ {𝐴})𝑧 → (𝑥(𝐹 ↾ {𝐴})𝑧𝐴(𝐹 ↾ {𝐴})𝑧)))
6261pm2.43d 53 . . . . . . . . . . . . . 14 (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (𝑥(𝐹 ↾ {𝐴})𝑧𝐴(𝐹 ↾ {𝐴})𝑧))
6362anim1d 604 . . . . . . . . . . . . 13 (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → (𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦)))
6463eximdv 2012 . . . . . . . . . . . 12 (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → ∃𝑧(𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦)))
6564exlimdv 2028 . . . . . . . . . . 11 (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (∃𝑥𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → ∃𝑧(𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦)))
6643, 65mpan9 502 . . . . . . . . . 10 ((¬ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → ∃𝑧(𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
6719eleq2i 2836 . . . . . . . . . . . . 13 (𝑦 ∈ ((𝐹 ↾ {𝐴}) “ {𝐴}) ↔ 𝑦 ∈ (𝐹 “ {𝐴}))
68 elimasni 5676 . . . . . . . . . . . . 13 (𝑦 ∈ ((𝐹 ↾ {𝐴}) “ {𝐴}) → 𝐴(𝐹 ↾ {𝐴})𝑦)
6967, 68sylbir 226 . . . . . . . . . . . 12 (𝑦 ∈ (𝐹 “ {𝐴}) → 𝐴(𝐹 ↾ {𝐴})𝑦)
70 vex 3353 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
7170, 52uniop 5138 . . . . . . . . . . . . . . . 16 𝑦, 𝑧⟩ = {𝑦, 𝑧}
72 opex 5090 . . . . . . . . . . . . . . . . . . 19 𝑦, 𝑧⟩ ∈ V
7372unisn 4612 . . . . . . . . . . . . . . . . . 18 {⟨𝑦, 𝑧⟩} = ⟨𝑦, 𝑧
7427brrelex1i 5330 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴(𝐹 ↾ {𝐴})𝑧𝐴 ∈ V)
75 brcnvg 5473 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∈ V ∧ 𝐴 ∈ V) → (𝑦(𝐹 ↾ {𝐴})𝐴𝐴(𝐹 ↾ {𝐴})𝑦))
7670, 74, 75sylancr 581 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴(𝐹 ↾ {𝐴})𝑧 → (𝑦(𝐹 ↾ {𝐴})𝐴𝐴(𝐹 ↾ {𝐴})𝑦))
7776biimpar 469 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴(𝐹 ↾ {𝐴})𝑧𝐴(𝐹 ↾ {𝐴})𝑦) → 𝑦(𝐹 ↾ {𝐴})𝐴)
7874adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦(𝐹 ↾ {𝐴})𝐴𝐴(𝐹 ↾ {𝐴})𝑧) → 𝐴 ∈ V)
79 breq2 4815 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = 𝐴 → (𝑦(𝐹 ↾ {𝐴})𝑥𝑦(𝐹 ↾ {𝐴})𝐴))
80 breq1 4814 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = 𝐴 → (𝑥(𝐹 ↾ {𝐴})𝑧𝐴(𝐹 ↾ {𝐴})𝑧))
8179, 80anbi12d 624 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝐴 → ((𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧) ↔ (𝑦(𝐹 ↾ {𝐴})𝐴𝐴(𝐹 ↾ {𝐴})𝑧)))
8281rspcev 3462 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ V ∧ (𝑦(𝐹 ↾ {𝐴})𝐴𝐴(𝐹 ↾ {𝐴})𝑧)) → ∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧))
8378, 82mpancom 679 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦(𝐹 ↾ {𝐴})𝐴𝐴(𝐹 ↾ {𝐴})𝑧) → ∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧))
8483ancoms 450 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴(𝐹 ↾ {𝐴})𝑧𝑦(𝐹 ↾ {𝐴})𝐴) → ∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧))
8577, 84syldan 585 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴(𝐹 ↾ {𝐴})𝑧𝐴(𝐹 ↾ {𝐴})𝑦) → ∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧))
8685anim1i 608 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴(𝐹 ↾ {𝐴})𝑧𝐴(𝐹 ↾ {𝐴})𝑦) ∧ ¬ 𝑧 = 𝑦) → (∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧) ∧ ¬ 𝑧 = 𝑦))
8786an32s 642 . . . . . . . . . . . . . . . . . . . 20 (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → (∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧) ∧ ¬ 𝑧 = 𝑦))
88 eldif 3744 . . . . . . . . . . . . . . . . . . . . 21 (⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) ↔ (⟨𝑦, 𝑧⟩ ∈ ((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∧ ¬ ⟨𝑦, 𝑧⟩ ∈ I ))
89 rexv 3373 . . . . . . . . . . . . . . . . . . . . . . 23 (∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧) ↔ ∃𝑥(𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧))
9070, 52brco 5463 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴}))𝑧 ↔ ∃𝑥(𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧))
91 df-br 4812 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴}))𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})))
9289, 90, 913bitr2ri 291 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝑦, 𝑧⟩ ∈ ((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ↔ ∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧))
9352ideq 5445 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 I 𝑧𝑦 = 𝑧)
94 df-br 4812 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 I 𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ I )
95 equcom 2115 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑧𝑧 = 𝑦)
9693, 94, 953bitr3i 292 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝑦, 𝑧⟩ ∈ I ↔ 𝑧 = 𝑦)
9796notbii 311 . . . . . . . . . . . . . . . . . . . . . 22 (¬ ⟨𝑦, 𝑧⟩ ∈ I ↔ ¬ 𝑧 = 𝑦)
9892, 97anbi12i 620 . . . . . . . . . . . . . . . . . . . . 21 ((⟨𝑦, 𝑧⟩ ∈ ((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∧ ¬ ⟨𝑦, 𝑧⟩ ∈ I ) ↔ (∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧) ∧ ¬ 𝑧 = 𝑦))
9988, 98bitr2i 267 . . . . . . . . . . . . . . . . . . . 20 ((∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧) ∧ ¬ 𝑧 = 𝑦) ↔ ⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
10087, 99sylib 209 . . . . . . . . . . . . . . . . . . 19 (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → ⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
101 snssi 4495 . . . . . . . . . . . . . . . . . . 19 (⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) → {⟨𝑦, 𝑧⟩} ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
102 uniss 4619 . . . . . . . . . . . . . . . . . . 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 3812 . . . . . . . . . . . . . . . . 17 (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → ⟨𝑦, 𝑧⟩ ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
105104unissd 4622 . . . . . . . . . . . . . . . 16 (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → 𝑦, 𝑧⟩ ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
10671, 105syl5eqssr 3812 . . . . . . . . . . . . . . 15 (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → {𝑦, 𝑧} ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
10770, 52prss 4507 . . . . . . . . . . . . . . 15 ((𝑦 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) ∧ 𝑧 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) ↔ {𝑦, 𝑧} ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
108106, 107sylibr 225 . . . . . . . . . . . . . 14 (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → (𝑦 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) ∧ 𝑧 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
109108simpld 488 . . . . . . . . . . . . 13 (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → 𝑦 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
110109ex 401 . . . . . . . . . . . 12 ((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → (𝐴(𝐹 ↾ {𝐴})𝑦𝑦 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
11169, 110syl5 34 . . . . . . . . . . 11 ((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → (𝑦 ∈ (𝐹 “ {𝐴}) → 𝑦 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
112111exlimiv 2025 . . . . . . . . . 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 3769 . . . . . . . 8 ((¬ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → (𝐹 “ {𝐴}) ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
115 ssdif0 4108 . . . . . . . 8 ((𝐹 “ {𝐴}) ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) ↔ ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ∅)
116114, 115sylib 209 . . . . . . 7 ((¬ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ∅)
117116ex 401 . . . . . 6 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ∅))
118 ndmima 5686 . . . . . . . . 9 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴}) “ {𝐴}) = ∅)
11919, 118syl5eqr 2813 . . . . . . . 8 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (𝐹 “ {𝐴}) = ∅)
120119difeq1d 3891 . . . . . . 7 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = (∅ ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
121 0dif 4141 . . . . . . 7 (∅ ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ∅
122120, 121syl6eq 2815 . . . . . 6 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ∅)
123117, 122pm2.61d1 172 . . . . 5 (¬ Fun (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ∅)
124123unieqd 4606 . . . 4 (¬ Fun (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ∅)
125124, 14syl6eq 2815 . . 3 (¬ Fun (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ∅)
12626, 125eqtr4d 2802 . 2 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹𝐴) = ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
12725, 126pm2.61i 176 1 (𝐹𝐴) = ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wal 1650   = wceq 1652  wex 1874  wcel 2155  wrex 3056  Vcvv 3350  cdif 3731  wss 3734  c0 4081  {csn 4336  {cpr 4338  cop 4342   cuni 4596   class class class wbr 4811   I cid 5186  ccnv 5278  dom cdm 5279  cres 5281  cima 5282  ccom 5283  Rel wrel 5284  Fun wfun 6064  cfv 6070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-fv 6078
This theorem is referenced by: (None)
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