Theorem fvopab6 7024

Description: Value of a function given by ordered-pair class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)

Hypotheses

Ref	Expression
fvopab6.1	⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐵)}
fvopab6.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
fvopab6.3	⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)

Assertion

Ref	Expression
fvopab6	⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅 ∧ 𝜓) → (𝐹‘𝐴) = 𝐶)

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

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

Proof of Theorem fvopab6

Step	Hyp	Ref	Expression
1		elex 3475	. . 3 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ V)
2		fvopab6.2	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3		fvopab6.3	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
4	3	eqeq2d 2773	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐶))
5	2, 4	anbi12d 643	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝑦 = 𝐵) ↔ (𝜓 ∧ 𝑦 = 𝐶)))
6		iba 536	. . . . 5 ⊢ (𝑦 = 𝐶 → (𝜓 ↔ (𝜓 ∧ 𝑦 = 𝐶)))
7	6	bicomd 226	. . . 4 ⊢ (𝑦 = 𝐶 → ((𝜓 ∧ 𝑦 = 𝐶) ↔ 𝜓))
8		moeq 3669	. . . . . 6 ⊢ ∃*𝑦 𝑦 = 𝐵
9	8	moani 2580	. . . . 5 ⊢ ∃*𝑦(𝜑 ∧ 𝑦 = 𝐵)
10	9	a1i 11	. . . 4 ⊢ (𝑥 ∈ V → ∃*𝑦(𝜑 ∧ 𝑦 = 𝐵))
11		fvopab6.1	. . . . 5 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐵)}
12		vex 3458	. . . . . . 7 ⊢ 𝑥 ∈ V
13	12	biantrur 539	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵)))
14	13	opabbii 5177	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵))}
15	11, 14	eqtri 2785	. . . 4 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵))}
16	5, 7, 10, 15	fvopab3ig 6985	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ 𝑅) → (𝜓 → (𝐹‘𝐴) = 𝐶))
17	1, 16	sylan 591	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅) → (𝜓 → (𝐹‘𝐴) = 𝐶))
18	17	3impia 1134	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅 ∧ 𝜓) → (𝐹‘𝐴) = 𝐶)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ∃*wmo 2564  Vcvv 3454  {copab 5172  ‘cfv 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6492  df-fun 6538  df-fv 6544

This theorem is used by: (None)