Theorem fvopab3ig 3784

Description: Value of a function given by ordered-pair class 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 groups:   x,y,A   x,B,y   x,C,y   χ,x,y

Proof of Theorem fvopab3ig

Step	Hyp	Ref	Expression
1		eleq1 1537	. . . . . . . . 9 ⊢ (x = A → (x ∈ C ↔ A ∈ C))
2		fvopab3ig.1	. . . . . . . . 9 ⊢ (x = A → (φ ↔ ψ))
3	1, 2	anbi12d 630	. . . . . . . 8 ⊢ (x = A → ((x ∈ C ⋀ φ) ↔ (A ∈ C ⋀ ψ)))
4		fvopab3ig.2	. . . . . . . . 9 ⊢ (y = B → (ψ ↔ χ))
5	4	anbi2d 618	. . . . . . . 8 ⊢ (y = B → ((A ∈ C ⋀ ψ) ↔ (A ∈ C ⋀ χ)))
6	3, 5	opelopabg 2823	. . . . . . 7 ⊢ ((A ∈ C ⋀ B ∈ D) → (⟨A, B⟩ ∈ {⟨x, y⟩∣(x ∈ C ⋀ φ)} ↔ (A ∈ C ⋀ χ)))
7	6	biimpar 419	. . . . . 6 ⊢ (((A ∈ C ⋀ B ∈ D) ⋀ (A ∈ C ⋀ χ)) → ⟨A, B⟩ ∈ {⟨x, y⟩∣(x ∈ C ⋀ φ)})
8	7	exp43 386	. . . . 5 ⊢ (A ∈ C → (B ∈ D → (A ∈ C → (χ → ⟨A, B⟩ ∈ {⟨x, y⟩∣(x ∈ C ⋀ φ)}))))
9	8	pm2.43a 66	. . . 4 ⊢ (A ∈ C → (B ∈ D → (χ → ⟨A, B⟩ ∈ {⟨x, y⟩∣(x ∈ C ⋀ φ)})))
10	9	imp 350	. . 3 ⊢ ((A ∈ C ⋀ B ∈ D) → (χ → ⟨A, B⟩ ∈ {⟨x, y⟩∣(x ∈ C ⋀ φ)}))
11		funopab 3554	. . . . . 6 ⊢ (Fun {⟨x, y⟩∣(x ∈ C ⋀ φ)} ↔ ∀x∃*y(x ∈ C ⋀ φ))
12		fvopab3ig.3	. . . . . . 7 ⊢ (x ∈ C → ∃*yφ)
13		moanimv 1431	. . . . . . 7 ⊢ (∃*y(x ∈ C ⋀ φ) ↔ (x ∈ C → ∃*yφ))
14	12, 13	mpbir 190	. . . . . 6 ⊢ ∃*y(x ∈ C ⋀ φ)
15	11, 14	mpgbir 990	. . . . 5 ⊢ Fun {⟨x, y⟩∣(x ∈ C ⋀ φ)}
16		funopfvg 3758	. . . . 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 698	. . . 4 ⊢ (B ∈ D → (⟨A, B⟩ ∈ {⟨x, y⟩∣(x ∈ C ⋀ φ)} → ({⟨x, y⟩∣(x ∈ C ⋀ φ)} ‘A) = B))
18	17	adantl 390	. . 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 3731	. . 3 ⊢ (F ‘A) = ({⟨x, y⟩∣(x ∈ C ⋀ φ)} ‘A)
22	21	eqeq1i 1485	. 2 ⊢ ((F ‘A) = B ↔ ({⟨x, y⟩∣(x ∈ C ⋀ φ)} ‘A) = B)
23	19, 22	syl6ibr 213	1 ⊢ ((A ∈ C ⋀ B ∈ D) → (χ → (F ‘A) = B))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 958   ∈ wcel 960  ∃*wmo 1383  ⟨cop 2415  {copab 2671  Fun wfun 3182   ‘cfv 3188

This theorem is referenced by:  fvopab4g 3785  oprabval6g 4038

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204

Copyright terms: Public domain