Theorem fvopab3ig 5388

Description: Value of a function given by ordered-pair class abstraction. (Contributed by set.mm contributors, 23-Oct-1999.)

Hypotheses

Ref	Expression
fvopab3ig.1	⊢ (x = A → (φ ↔ ψ))
fvopab3ig.2	⊢ (y = B → (ψ ↔ χ))
fvopab3ig.3	⊢ (x ∈ C → ∃*yφ)
fvopab3ig.4	⊢ F = {⟨x, y⟩ ∣ (x ∈ C ∧ φ)}

Assertion

Ref	Expression
fvopab3ig	⊢ ((A ∈ C ∧ B ∈ D) → (χ → (F ‘A) = B))

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

Allowed substitution hints:   φ(x,y)   ψ(x,y)   D(x,y)   F(x,y)

Proof of Theorem fvopab3ig

Step	Hyp	Ref	Expression
1		funopab 5140	. . . 4 ⊢ (Fun {⟨x, y⟩ ∣ (x ∈ C ∧ φ)} ↔ ∀x∃*y(x ∈ C ∧ φ))
2		fvopab3ig.3	. . . . 5 ⊢ (x ∈ C → ∃*yφ)
3		moanimv 2262	. . . . 5 ⊢ (∃*y(x ∈ C ∧ φ) ↔ (x ∈ C → ∃*yφ))
4	2, 3	mpbir 200	. . . 4 ⊢ ∃*y(x ∈ C ∧ φ)
5	1, 4	mpgbir 1550	. . 3 ⊢ Fun {⟨x, y⟩ ∣ (x ∈ C ∧ φ)}
6		simpl 443	. . . 4 ⊢ ((A ∈ C ∧ B ∈ D) → A ∈ C)
7		eleq1 2413	. . . . . . 7 ⊢ (x = A → (x ∈ C ↔ A ∈ C))
8		fvopab3ig.1	. . . . . . 7 ⊢ (x = A → (φ ↔ ψ))
9	7, 8	anbi12d 691	. . . . . 6 ⊢ (x = A → ((x ∈ C ∧ φ) ↔ (A ∈ C ∧ ψ)))
10		fvopab3ig.2	. . . . . . 7 ⊢ (y = B → (ψ ↔ χ))
11	10	anbi2d 684	. . . . . 6 ⊢ (y = B → ((A ∈ C ∧ ψ) ↔ (A ∈ C ∧ χ)))
12	9, 11	opelopabg 4706	. . . . 5 ⊢ ((A ∈ C ∧ B ∈ D) → (⟨A, B⟩ ∈ {⟨x, y⟩ ∣ (x ∈ C ∧ φ)} ↔ (A ∈ C ∧ χ)))
13	12	biimprd 214	. . . 4 ⊢ ((A ∈ C ∧ B ∈ D) → ((A ∈ C ∧ χ) → ⟨A, B⟩ ∈ {⟨x, y⟩ ∣ (x ∈ C ∧ φ)}))
14	6, 13	mpand 656	. . 3 ⊢ ((A ∈ C ∧ B ∈ D) → (χ → ⟨A, B⟩ ∈ {⟨x, y⟩ ∣ (x ∈ C ∧ φ)}))
15		funopfv 5358	. . 3 ⊢ (Fun {⟨x, y⟩ ∣ (x ∈ C ∧ φ)} → (⟨A, B⟩ ∈ {⟨x, y⟩ ∣ (x ∈ C ∧ φ)} → ({⟨x, y⟩ ∣ (x ∈ C ∧ φ)} ‘A) = B))
16	5, 14, 15	ee02 1377	. 2 ⊢ ((A ∈ C ∧ B ∈ D) → (χ → ({⟨x, y⟩ ∣ (x ∈ C ∧ φ)} ‘A) = B))
17		fvopab3ig.4	. . . 4 ⊢ F = {⟨x, y⟩ ∣ (x ∈ C ∧ φ)}
18	17	fveq1i 5330	. . 3 ⊢ (F ‘A) = ({⟨x, y⟩ ∣ (x ∈ C ∧ φ)} ‘A)
19	18	eqeq1i 2360	. 2 ⊢ ((F ‘A) = B ↔ ({⟨x, y⟩ ∣ (x ∈ C ∧ φ)} ‘A) = B)
20	16, 19	syl6ibr 218	1 ⊢ ((A ∈ C ∧ B ∈ D) → (χ → (F ‘A) = B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  ⟨cop 4562  {copab 4623  Fun wfun 4776   ‘cfv 4782

This theorem was proved from 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 theorem 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 referenced by:  fvopab4g  5389  ov6g  5601