Theorem fvopab4t 5386

Description: Closed theorem form of fvopab4 5390. (Contributed by set.mm contributors, 21-Feb-2013.)

Assertion

Ref	Expression
fvopab4t	⊢ ((∀x∀y(x = A → B = C) ∧ ∀x F = {⟨x, y⟩ ∣ (x ∈ D ∧ y = B)} ∧ (A ∈ D ∧ C ∈ V)) → (F ‘A) = C)

Distinct variable groups:   x,y,A   y,B   x,C,y   x,D,y

Allowed substitution hints:   B(x)   F(x, y)   V(x, y)

Proof of Theorem fvopab4t

Step	Hyp	Ref	Expression
1		elex 2868	. . 3 ⊢ (C ∈ V → C ∈ V)
2	1	anim2i 552	. 2 ⊢ ((A ∈ D ∧ C ∈ V) → (A ∈ D ∧ C ∈ V))
3		funopab4 5142	. . . 4 ⊢ Fun {⟨x, y⟩ ∣ (x ∈ D ∧ y = B)}
4		simp2 956	. . . . . 6 ⊢ ((∀x∀y(x = A → B = C) ∧ ∀x F = {⟨x, y⟩ ∣ (x ∈ D ∧ y = B)} ∧ (A ∈ D ∧ C ∈ V)) → ∀x F = {⟨x, y⟩ ∣ (x ∈ D ∧ y = B)})
5	4	19.21bi 1758	. . . . 5 ⊢ ((∀x∀y(x = A → B = C) ∧ ∀x F = {⟨x, y⟩ ∣ (x ∈ D ∧ y = B)} ∧ (A ∈ D ∧ C ∈ V)) → F = {⟨x, y⟩ ∣ (x ∈ D ∧ y = B)})
6	5	funeqd 5130	. . . 4 ⊢ ((∀x∀y(x = A → B = C) ∧ ∀x F = {⟨x, y⟩ ∣ (x ∈ D ∧ y = B)} ∧ (A ∈ D ∧ C ∈ V)) → (Fun F ↔ Fun {⟨x, y⟩ ∣ (x ∈ D ∧ y = B)}))
7	3, 6	mpbiri 224	. . 3 ⊢ ((∀x∀y(x = A → B = C) ∧ ∀x F = {⟨x, y⟩ ∣ (x ∈ D ∧ y = B)} ∧ (A ∈ D ∧ C ∈ V)) → Fun F)
8		simp3l 983	. . . . 5 ⊢ ((∀x∀y(x = A → B = C) ∧ ∀x F = {⟨x, y⟩ ∣ (x ∈ D ∧ y = B)} ∧ (A ∈ D ∧ C ∈ V)) → A ∈ D)
9		eqidd 2354	. . . . 5 ⊢ ((∀x∀y(x = A → B = C) ∧ ∀x F = {⟨x, y⟩ ∣ (x ∈ D ∧ y = B)} ∧ (A ∈ D ∧ C ∈ V)) → C = C)
10		eleq1 2413	. . . . . . . . . . 11 ⊢ (x = A → (x ∈ D ↔ A ∈ D))
11		eqeq2 2362	. . . . . . . . . . 11 ⊢ (B = C → (y = B ↔ y = C))
12	10, 11	bi2anan9 843	. . . . . . . . . 10 ⊢ ((x = A ∧ B = C) → ((x ∈ D ∧ y = B) ↔ (A ∈ D ∧ y = C)))
13	12	ex 423	. . . . . . . . 9 ⊢ (x = A → (B = C → ((x ∈ D ∧ y = B) ↔ (A ∈ D ∧ y = C))))
14	13	a2i 12	. . . . . . . 8 ⊢ ((x = A → B = C) → (x = A → ((x ∈ D ∧ y = B) ↔ (A ∈ D ∧ y = C))))
15	14	2alimi 1560	. . . . . . 7 ⊢ (∀x∀y(x = A → B = C) → ∀x∀y(x = A → ((x ∈ D ∧ y = B) ↔ (A ∈ D ∧ y = C))))
16	15	3ad2ant1 976	. . . . . 6 ⊢ ((∀x∀y(x = A → B = C) ∧ ∀x F = {⟨x, y⟩ ∣ (x ∈ D ∧ y = B)} ∧ (A ∈ D ∧ C ∈ V)) → ∀x∀y(x = A → ((x ∈ D ∧ y = B) ↔ (A ∈ D ∧ y = C))))
17		eqeq1 2359	. . . . . . . . 9 ⊢ (y = C → (y = C ↔ C = C))
18	17	anbi2d 684	. . . . . . . 8 ⊢ (y = C → ((A ∈ D ∧ y = C) ↔ (A ∈ D ∧ C = C)))
19	18	gen2 1547	. . . . . . 7 ⊢ ∀x∀y(y = C → ((A ∈ D ∧ y = C) ↔ (A ∈ D ∧ C = C)))
20	19	a1i 10	. . . . . 6 ⊢ ((∀x∀y(x = A → B = C) ∧ ∀x F = {⟨x, y⟩ ∣ (x ∈ D ∧ y = B)} ∧ (A ∈ D ∧ C ∈ V)) → ∀x∀y(y = C → ((A ∈ D ∧ y = C) ↔ (A ∈ D ∧ C = C))))
21		simp3 957	. . . . . 6 ⊢ ((∀x∀y(x = A → B = C) ∧ ∀x F = {⟨x, y⟩ ∣ (x ∈ D ∧ y = B)} ∧ (A ∈ D ∧ C ∈ V)) → (A ∈ D ∧ C ∈ V))
22		opelopabt 4700	. . . . . 6 ⊢ ((∀x∀y(x = A → ((x ∈ D ∧ y = B) ↔ (A ∈ D ∧ y = C))) ∧ ∀x∀y(y = C → ((A ∈ D ∧ y = C) ↔ (A ∈ D ∧ C = C))) ∧ (A ∈ D ∧ C ∈ V)) → (⟨A, C⟩ ∈ {⟨x, y⟩ ∣ (x ∈ D ∧ y = B)} ↔ (A ∈ D ∧ C = C)))
23	16, 20, 21, 22	syl3anc 1182	. . . . 5 ⊢ ((∀x∀y(x = A → B = C) ∧ ∀x F = {⟨x, y⟩ ∣ (x ∈ D ∧ y = B)} ∧ (A ∈ D ∧ C ∈ V)) → (⟨A, C⟩ ∈ {⟨x, y⟩ ∣ (x ∈ D ∧ y = B)} ↔ (A ∈ D ∧ C = C)))
24	8, 9, 23	mpbir2and 888	. . . 4 ⊢ ((∀x∀y(x = A → B = C) ∧ ∀x F = {⟨x, y⟩ ∣ (x ∈ D ∧ y = B)} ∧ (A ∈ D ∧ C ∈ V)) → ⟨A, C⟩ ∈ {⟨x, y⟩ ∣ (x ∈ D ∧ y = B)})
25	24, 5	eleqtrrd 2430	. . 3 ⊢ ((∀x∀y(x = A → B = C) ∧ ∀x F = {⟨x, y⟩ ∣ (x ∈ D ∧ y = B)} ∧ (A ∈ D ∧ C ∈ V)) → ⟨A, C⟩ ∈ F)
26		funopfv 5358	. . 3 ⊢ (Fun F → (⟨A, C⟩ ∈ F → (F ‘A) = C))
27	7, 25, 26	sylc 56	. 2 ⊢ ((∀x∀y(x = A → B = C) ∧ ∀x F = {⟨x, y⟩ ∣ (x ∈ D ∧ y = B)} ∧ (A ∈ D ∧ C ∈ V)) → (F ‘A) = C)
28	2, 27	syl3an3 1217	1 ⊢ ((∀x∀y(x = A → B = C) ∧ ∀x F = {⟨x, y⟩ ∣ (x ∈ D ∧ y = B)} ∧ (A ∈ D ∧ C ∈ V)) → (F ‘A) = C)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Vcvv 2860  ⟨cop 4562  {copab 4623  Fun wfun 4776   ‘cfv 4782

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fv 4796

This theorem is used by: (None)