Theorem fvopab3ig 5560

Description: Value of a function given by ordered-pair class abstraction. (Contributed by NM, 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		eleq1 2229	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶))
2		fvopab3ig.1	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	1, 2	anbi12d 465	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐶 ∧ 𝜑) ↔ (𝐴 ∈ 𝐶 ∧ 𝜓)))
4		fvopab3ig.2	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
5	4	anbi2d 460	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝐶 ∧ 𝜓) ↔ (𝐴 ∈ 𝐶 ∧ 𝜒)))
6	3, 5	opelopabg 4246	. . . . . 6 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)} ↔ (𝐴 ∈ 𝐶 ∧ 𝜒)))
7	6	biimpar 295	. . . . 5 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝜒)) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)})
8	7	exp43 370	. . . 4 ⊢ (𝐴 ∈ 𝐶 → (𝐵 ∈ 𝐷 → (𝐴 ∈ 𝐶 → (𝜒 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}))))
9	8	pm2.43a 51	. . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐵 ∈ 𝐷 → (𝜒 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)})))
10	9	imp 123	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝜒 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}))
11		fvopab3ig.4	. . . 4 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}
12	11	fveq1i 5487	. . 3 ⊢ (𝐹‘𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}‘𝐴)
13		funopab 5223	. . . . 5 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)} ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐶 ∧ 𝜑))
14		fvopab3ig.3	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 → ∃*𝑦𝜑)
15		moanimv 2089	. . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝐶 ∧ 𝜑) ↔ (𝑥 ∈ 𝐶 → ∃*𝑦𝜑))
16	14, 15	mpbir 145	. . . . 5 ⊢ ∃*𝑦(𝑥 ∈ 𝐶 ∧ 𝜑)
17	13, 16	mpgbir 1441	. . . 4 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}
18		funopfv 5526	. . . 4 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)} → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)} → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}‘𝐴) = 𝐵))
19	17, 18	ax-mp 5	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)} → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}‘𝐴) = 𝐵)
20	12, 19	syl5eq 2211	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)} → (𝐹‘𝐴) = 𝐵)
21	10, 20	syl6 33	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝜒 → (𝐹‘𝐴) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343  ∃*wmo 2015   ∈ wcel 2136  ⟨cop 3579  {copab 4042  Fun wfun 5182  ‘cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196

This theorem is referenced by:  fvmptg  5562  fvopab6  5582  ov6g  5979