Theorem fvopab3ig 2869

Description: Value of a function given by ordered pair abstraction.

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 group(s):   x,y,A   x,B,y   x,C,y   χ,x,y

Proof of Theorem fvopab3ig

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . . . . . . 9 ⊢ (x = A → (x ∈ C ↔ A ∈ C))
2		fvopab3ig.1	. . . . . . . . 9 ⊢ (x = A → (φ ↔ ψ))
3	1, 2	anbi12d 476	. . . . . . . 8 ⊢ (x = A → ((x ∈ C ∧ φ) ↔ (A ∈ C ∧ ψ)))
4		fvopab3ig.2	. . . . . . . . 9 ⊢ (y = B → (ψ ↔ χ))
5	4	anbi2d 468	. . . . . . . 8 ⊢ (y = B → ((A ∈ C ∧ ψ) ↔ (A ∈ C ∧ χ)))
6	3, 5	opelopabg 2115	. . . . . . 7 ⊢ ((A ∈ C ∧ B ∈ D) → (⟨A, B⟩ ∈ {⟨x, y⟩∣(x ∈ C ∧ φ)} ↔ (A ∈ C ∧ χ)))
7	6	biimpar 325	. . . . . 6 ⊢ (((A ∈ C ∧ B ∈ D) ∧ (A ∈ C ∧ χ)) → ⟨A, B⟩ ∈ {⟨x, y⟩∣(x ∈ C ∧ φ)})
8	7	exp43 301	. . . . 5 ⊢ (A ∈ C → (B ∈ D → (A ∈ C → (χ → ⟨A, B⟩ ∈ {⟨x, y⟩∣(x ∈ C ∧ φ)}))))
9	8	pm2.43a 60	. . . 4 ⊢ (A ∈ C → (B ∈ D → (χ → ⟨A, B⟩ ∈ {⟨x, y⟩∣(x ∈ C ∧ φ)})))
10	9	imp 277	. . 3 ⊢ ((A ∈ C ∧ B ∈ D) → (χ → ⟨A, B⟩ ∈ {⟨x, y⟩∣(x ∈ C ∧ φ)}))
11		funopab 2694	. . . . . 6 ⊢ (Fun {⟨x, y⟩∣(x ∈ C ∧ φ)} ↔ ∀x∃*y(x ∈ C ∧ φ))
12		fvopab3ig.3	. . . . . . 7 ⊢ (x ∈ C → ∃*yφ)
13		moanimv 1052	. . . . . . 7 ⊢ (∃*y(x ∈ C ∧ φ) ↔ (x ∈ C → ∃*yφ))
14	12, 13	mpbir 165	. . . . . 6 ⊢ ∃*y(x ∈ C ∧ φ)
15	11, 14	mpgbir 686	. . . . 5 ⊢ Fun {⟨x, y⟩∣(x ∈ C ∧ φ)}
16		funopfvg 2854	. . . . 5 ⊢ ((B ∈ D ∧ Fun {⟨x, y⟩∣(x ∈ C ∧ φ)}) → (⟨A, B⟩ ∈ {⟨x, y⟩∣(x ∈ C ∧ φ)} → ({⟨x, y⟩∣(x ∈ C ∧ φ)} ‘A) = B))
17	15, 16	mpan2 519	. . . 4 ⊢ (B ∈ D → (⟨A, B⟩ ∈ {⟨x, y⟩∣(x ∈ C ∧ φ)} → ({⟨x, y⟩∣(x ∈ C ∧ φ)} ‘A) = B))
18	17	adantl 305	. . 3 ⊢ ((A ∈ C ∧ B ∈ D) → (⟨A, B⟩ ∈ {⟨x, y⟩∣(x ∈ C ∧ φ)} → ({⟨x, y⟩∣(x ∈ C ∧ φ)} ‘A) = B))
19	10, 18	syld 27	. 2 ⊢ ((A ∈ C ∧ B ∈ D) → (χ → ({⟨x, y⟩∣(x ∈ C ∧ φ)} ‘A) = B))
20		fvopab3ig.4	. . . 4 ⊢ F = {⟨x, y⟩∣(x ∈ C ∧ φ)}
21	20	fveq1i 2833	. . 3 ⊢ (F ‘A) = ({⟨x, y⟩∣(x ∈ C ∧ φ)} ‘A)
22	21	cleq1i 1108	. 2 ⊢ ((F ‘A) = B ↔ ({⟨x, y⟩∣(x ∈ C ∧ φ)} ‘A) = B)
23	19, 22	syl6ibr 186	1 ⊢ ((A ∈ C ∧ B ∈ D) → (χ → (F ‘A) = B))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃*wmo 1008   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  {copab 2055  Fun wfun 2416   ‘cfv 2422

This theorem is referenced by:  fvopab4g 2870

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438

metamath.org