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Theorem dffv2 6433
Description: Alternate definition of function value df-fv 6057 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 4352 . . . . 5 (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
2 fvres 6368 . . . . 5 (𝐴 ∈ {𝐴} → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹𝐴))
31, 2sylbi 207 . . . 4 (𝐴 ∈ V → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹𝐴))
4 fvprc 6346 . . . . 5 𝐴 ∈ V → ((𝐹 ↾ {𝐴})‘𝐴) = ∅)
5 fvprc 6346 . . . . 5 𝐴 ∈ V → (𝐹𝐴) = ∅)
64, 5eqtr4d 2797 . . . 4 𝐴 ∈ V → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹𝐴))
73, 6pm2.61i 176 . . 3 ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹𝐴)
8 funfv 6427 . . . 4 (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴})‘𝐴) = ((𝐹 ↾ {𝐴}) “ {𝐴}))
9 df-fun 6051 . . . . . . . . . . . . 13 (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ⊆ I ))
109simprbi 483 . . . . . . . . . . . 12 (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ⊆ I )
11 ssdif0 4085 . . . . . . . . . . . 12 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ⊆ I ↔ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) = ∅)
1210, 11sylib 208 . . . . . . . . . . 11 (Fun (𝐹 ↾ {𝐴}) → (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) = ∅)
1312unieqd 4598 . . . . . . . . . 10 (Fun (𝐹 ↾ {𝐴}) → (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) = ∅)
14 uni0 4617 . . . . . . . . . 10 ∅ = ∅
1513, 14syl6eq 2810 . . . . . . . . 9 (Fun (𝐹 ↾ {𝐴}) → (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) = ∅)
1615unieqd 4598 . . . . . . . 8 (Fun (𝐹 ↾ {𝐴}) → (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) = ∅)
1716, 14syl6eq 2810 . . . . . . 7 (Fun (𝐹 ↾ {𝐴}) → (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) = ∅)
1817difeq2d 3871 . . . . . 6 (Fun (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ((𝐹 “ {𝐴}) ∖ ∅))
19 resima 5589 . . . . . . 7 ((𝐹 ↾ {𝐴}) “ {𝐴}) = (𝐹 “ {𝐴})
20 dif0 4093 . . . . . . 7 ((𝐹 “ {𝐴}) ∖ ∅) = (𝐹 “ {𝐴})
2119, 20eqtr4i 2785 . . . . . 6 ((𝐹 ↾ {𝐴}) “ {𝐴}) = ((𝐹 “ {𝐴}) ∖ ∅)
2218, 21syl6reqr 2813 . . . . 5 (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴}) “ {𝐴}) = ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
2322unieqd 4598 . . . 4 (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴}) “ {𝐴}) = ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
248, 23eqtrd 2794 . . 3 (Fun (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴})‘𝐴) = ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
257, 24syl5eqr 2808 . 2 (Fun (𝐹 ↾ {𝐴}) → (𝐹𝐴) = ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
26 nfunsn 6386 . . 3 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹𝐴) = ∅)
27 relres 5584 . . . . . . . . . . . . . . 15 Rel (𝐹 ↾ {𝐴})
28 dffun3 6060 . . . . . . . . . . . . . . 15 (Fun (𝐹 ↾ {𝐴}) ↔ (Rel (𝐹 ↾ {𝐴}) ∧ ∀𝑥𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧𝑧 = 𝑦)))
2927, 28mpbiran 991 . . . . . . . . . . . . . 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 1896 . . . . . . . . . . . . . . . . . 18 (∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧𝑧 = 𝑦) ↔ ∀𝑧 ¬ (𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
32 alnex 1855 . . . . . . . . . . . . . . . . . 18 (∀𝑧 ¬ (𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ↔ ¬ ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
3331, 32bitri 264 . . . . . . . . . . . . . . . . 17 (∀𝑧(𝑥(𝐹 ↾ {𝐴})𝑧𝑧 = 𝑦) ↔ ¬ ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
3433exbii 1923 . . . . . . . . . . . . . . . 16 (∃𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧𝑧 = 𝑦) ↔ ∃𝑦 ¬ ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
35 exnal 1903 . . . . . . . . . . . . . . . 16 (∃𝑦 ¬ ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ↔ ¬ ∀𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
3634, 35bitri 264 . . . . . . . . . . . . . . 15 (∃𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧𝑧 = 𝑦) ↔ ¬ ∀𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
3736albii 1896 . . . . . . . . . . . . . 14 (∀𝑥𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧𝑧 = 𝑦) ↔ ∀𝑥 ¬ ∀𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
38 alnex 1855 . . . . . . . . . . . . . 14 (∀𝑥 ¬ ∀𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ↔ ¬ ∃𝑥𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
3929, 37, 383bitrri 287 . . . . . . . . . . . . 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 2200 . . . . . . . . . . . . 13 (∀𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → ∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
4241eximi 1911 . . . . . . . . . . . 12 (∃𝑥𝑦𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → ∃𝑥𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
4340, 42sylbi 207 . . . . . . . . . . 11 (¬ Fun (𝐹 ↾ {𝐴}) → ∃𝑥𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
44 snssi 4484 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → {𝐴} ⊆ dom (𝐹 ↾ {𝐴}))
45 residm 5588 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = (𝐹 ↾ {𝐴})
4645dmeqi 5480 . . . . . . . . . . . . . . . . . . . . 21 dom ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = dom (𝐹 ↾ {𝐴})
47 ssdmres 5578 . . . . . . . . . . . . . . . . . . . . . 22 ({𝐴} ⊆ dom (𝐹 ↾ {𝐴}) ↔ dom ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = {𝐴})
4847biimpi 206 . . . . . . . . . . . . . . . . . . . . 21 ({𝐴} ⊆ dom (𝐹 ↾ {𝐴}) → dom ((𝐹 ↾ {𝐴}) ↾ {𝐴}) = {𝐴})
4946, 48syl5eqr 2808 . . . . . . . . . . . . . . . . . . . 20 ({𝐴} ⊆ dom (𝐹 ↾ {𝐴}) → dom (𝐹 ↾ {𝐴}) = {𝐴})
5044, 49syl 17 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → dom (𝐹 ↾ {𝐴}) = {𝐴})
51 vex 3343 . . . . . . . . . . . . . . . . . . . 20 𝑥 ∈ V
52 vex 3343 . . . . . . . . . . . . . . . . . . . 20 𝑧 ∈ V
5351, 52breldm 5484 . . . . . . . . . . . . . . . . . . 19 (𝑥(𝐹 ↾ {𝐴})𝑧𝑥 ∈ dom (𝐹 ↾ {𝐴}))
54 eleq2 2828 . . . . . . . . . . . . . . . . . . . . 21 (dom (𝐹 ↾ {𝐴}) = {𝐴} → (𝑥 ∈ dom (𝐹 ↾ {𝐴}) ↔ 𝑥 ∈ {𝐴}))
55 velsn 4337 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
5654, 55syl6bb 276 . . . . . . . . . . . . . . . . . . . 20 (dom (𝐹 ↾ {𝐴}) = {𝐴} → (𝑥 ∈ dom (𝐹 ↾ {𝐴}) ↔ 𝑥 = 𝐴))
5756biimpa 502 . . . . . . . . . . . . . . . . . . 19 ((dom (𝐹 ↾ {𝐴}) = {𝐴} ∧ 𝑥 ∈ dom (𝐹 ↾ {𝐴})) → 𝑥 = 𝐴)
5850, 53, 57syl2an 495 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ dom (𝐹 ↾ {𝐴}) ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) → 𝑥 = 𝐴)
5958breq1d 4814 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ dom (𝐹 ↾ {𝐴}) ∧ 𝑥(𝐹 ↾ {𝐴})𝑧) → (𝑥(𝐹 ↾ {𝐴})𝑧𝐴(𝐹 ↾ {𝐴})𝑧))
6059biimpd 219 . . . . . . . . . . . . . . . 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 53 . . . . . . . . . . . . . 14 (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (𝑥(𝐹 ↾ {𝐴})𝑧𝐴(𝐹 ↾ {𝐴})𝑧))
6362anim1d 589 . . . . . . . . . . . . 13 (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → (𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦)))
6463eximdv 1995 . . . . . . . . . . . 12 (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (∃𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → ∃𝑧(𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦)))
6564exlimdv 2010 . . . . . . . . . . 11 (𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (∃𝑥𝑧(𝑥(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → ∃𝑧(𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦)))
6643, 65mpan9 487 . . . . . . . . . 10 ((¬ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → ∃𝑧(𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦))
6719eleq2i 2831 . . . . . . . . . . . . 13 (𝑦 ∈ ((𝐹 ↾ {𝐴}) “ {𝐴}) ↔ 𝑦 ∈ (𝐹 “ {𝐴}))
68 elimasni 5650 . . . . . . . . . . . . 13 (𝑦 ∈ ((𝐹 ↾ {𝐴}) “ {𝐴}) → 𝐴(𝐹 ↾ {𝐴})𝑦)
6967, 68sylbir 225 . . . . . . . . . . . 12 (𝑦 ∈ (𝐹 “ {𝐴}) → 𝐴(𝐹 ↾ {𝐴})𝑦)
70 vex 3343 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
7170, 52uniop 5125 . . . . . . . . . . . . . . . 16 𝑦, 𝑧⟩ = {𝑦, 𝑧}
72 opex 5081 . . . . . . . . . . . . . . . . . . 19 𝑦, 𝑧⟩ ∈ V
7372unisn 4603 . . . . . . . . . . . . . . . . . 18 {⟨𝑦, 𝑧⟩} = ⟨𝑦, 𝑧
7427brrelexi 5315 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴(𝐹 ↾ {𝐴})𝑧𝐴 ∈ V)
75 brcnvg 5458 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∈ V ∧ 𝐴 ∈ V) → (𝑦(𝐹 ↾ {𝐴})𝐴𝐴(𝐹 ↾ {𝐴})𝑦))
7670, 74, 75sylancr 698 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴(𝐹 ↾ {𝐴})𝑧 → (𝑦(𝐹 ↾ {𝐴})𝐴𝐴(𝐹 ↾ {𝐴})𝑦))
7776biimpar 503 . . . . . . . . . . . . . . . . . . . . . . 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 4808 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = 𝐴 → (𝑦(𝐹 ↾ {𝐴})𝑥𝑦(𝐹 ↾ {𝐴})𝐴))
80 breq1 4807 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = 𝐴 → (𝑥(𝐹 ↾ {𝐴})𝑧𝐴(𝐹 ↾ {𝐴})𝑧))
8179, 80anbi12d 749 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝐴 → ((𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧) ↔ (𝑦(𝐹 ↾ {𝐴})𝐴𝐴(𝐹 ↾ {𝐴})𝑧)))
8281rspcev 3449 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ V ∧ (𝑦(𝐹 ↾ {𝐴})𝐴𝐴(𝐹 ↾ {𝐴})𝑧)) → ∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧))
8378, 82mpancom 706 . . . . . . . . . . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴(𝐹 ↾ {𝐴})𝑧𝐴(𝐹 ↾ {𝐴})𝑦) → ∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧))
8685anim1i 593 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴(𝐹 ↾ {𝐴})𝑧𝐴(𝐹 ↾ {𝐴})𝑦) ∧ ¬ 𝑧 = 𝑦) → (∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧) ∧ ¬ 𝑧 = 𝑦))
8786an32s 881 . . . . . . . . . . . . . . . . . . . 20 (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → (∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧) ∧ ¬ 𝑧 = 𝑦))
88 eldif 3725 . . . . . . . . . . . . . . . . . . . . 21 (⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) ↔ (⟨𝑦, 𝑧⟩ ∈ ((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∧ ¬ ⟨𝑦, 𝑧⟩ ∈ I ))
89 rexv 3360 . . . . . . . . . . . . . . . . . . . . . . 23 (∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧) ↔ ∃𝑥(𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧))
9070, 52brco 5448 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴}))𝑧 ↔ ∃𝑥(𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧))
91 df-br 4805 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴}))𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ ((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})))
9289, 90, 913bitr2ri 289 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝑦, 𝑧⟩ ∈ ((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ↔ ∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧))
9352ideq 5430 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 I 𝑧𝑦 = 𝑧)
94 df-br 4805 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 I 𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ I )
95 equcom 2100 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑧𝑧 = 𝑦)
9693, 94, 953bitr3i 290 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝑦, 𝑧⟩ ∈ I ↔ 𝑧 = 𝑦)
9796notbii 309 . . . . . . . . . . . . . . . . . . . . . 22 (¬ ⟨𝑦, 𝑧⟩ ∈ I ↔ ¬ 𝑧 = 𝑦)
9892, 97anbi12i 735 . . . . . . . . . . . . . . . . . . . . 21 ((⟨𝑦, 𝑧⟩ ∈ ((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∧ ¬ ⟨𝑦, 𝑧⟩ ∈ I ) ↔ (∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧) ∧ ¬ 𝑧 = 𝑦))
9988, 98bitr2i 265 . . . . . . . . . . . . . . . . . . . 20 ((∃𝑥 ∈ V (𝑦(𝐹 ↾ {𝐴})𝑥𝑥(𝐹 ↾ {𝐴})𝑧) ∧ ¬ 𝑧 = 𝑦) ↔ ⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
10087, 99sylib 208 . . . . . . . . . . . . . . . . . . 19 (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → ⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
101 snssi 4484 . . . . . . . . . . . . . . . . . . 19 (⟨𝑦, 𝑧⟩ ∈ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) → {⟨𝑦, 𝑧⟩} ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
102 uniss 4610 . . . . . . . . . . . . . . . . . . 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 3791 . . . . . . . . . . . . . . . . 17 (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → ⟨𝑦, 𝑧⟩ ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
105104unissd 4614 . . . . . . . . . . . . . . . 16 (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → 𝑦, 𝑧⟩ ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
10671, 105syl5eqssr 3791 . . . . . . . . . . . . . . 15 (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → {𝑦, 𝑧} ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
10770, 52prss 4496 . . . . . . . . . . . . . . 15 ((𝑦 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) ∧ 𝑧 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) ↔ {𝑦, 𝑧} ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
108106, 107sylibr 224 . . . . . . . . . . . . . 14 (((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) ∧ 𝐴(𝐹 ↾ {𝐴})𝑦) → (𝑦 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) ∧ 𝑧 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
109108simpld 477 . . . . . . . . . . . . 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 34 . . . . . . . . . . 11 ((𝐴(𝐹 ↾ {𝐴})𝑧 ∧ ¬ 𝑧 = 𝑦) → (𝑦 ∈ (𝐹 “ {𝐴}) → 𝑦 (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
112111exlimiv 2007 . . . . . . . . . 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 3750 . . . . . . . 8 ((¬ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → (𝐹 “ {𝐴}) ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
115 ssdif0 4085 . . . . . . . 8 ((𝐹 “ {𝐴}) ⊆ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ) ↔ ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ∅)
116114, 115sylib 208 . . . . . . 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 5660 . . . . . . . . 9 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 ↾ {𝐴}) “ {𝐴}) = ∅)
11919, 118syl5eqr 2808 . . . . . . . 8 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → (𝐹 “ {𝐴}) = ∅)
120119difeq1d 3870 . . . . . . 7 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = (∅ ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )))
121 0dif 4120 . . . . . . 7 (∅ ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ∅
122120, 121syl6eq 2810 . . . . . 6 𝐴 ∈ dom (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ∅)
123117, 122pm2.61d1 171 . . . . 5 (¬ Fun (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ∅)
124123unieqd 4598 . . . 4 (¬ Fun (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ∅)
125124, 14syl6eq 2810 . . 3 (¬ Fun (𝐹 ↾ {𝐴}) → ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I )) = ∅)
12626, 125eqtr4d 2797 . 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 196  wa 383  wal 1630   = wceq 1632  wex 1853  wcel 2139  wrex 3051  Vcvv 3340  cdif 3712  wss 3715  c0 4058  {csn 4321  {cpr 4323  cop 4327   cuni 4588   class class class wbr 4804   I cid 5173  ccnv 5265  dom cdm 5266  cres 5268  cima 5269  ccom 5270  Rel wrel 5271  Fun wfun 6043  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057
This theorem is referenced by: (None)
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