Theorem fvopab6 7031

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 3492	. . 3 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ V)
2		fvopab6.2	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3		fvopab6.3	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
4	3	eqeq2d 2742	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐶))
5	2, 4	anbi12d 630	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝑦 = 𝐵) ↔ (𝜓 ∧ 𝑦 = 𝐶)))
6		iba 527	. . . . 5 ⊢ (𝑦 = 𝐶 → (𝜓 ↔ (𝜓 ∧ 𝑦 = 𝐶)))
7	6	bicomd 222	. . . 4 ⊢ (𝑦 = 𝐶 → ((𝜓 ∧ 𝑦 = 𝐶) ↔ 𝜓))
8		moeq 3703	. . . . . 6 ⊢ ∃*𝑦 𝑦 = 𝐵
9	8	moani 2546	. . . . 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 3477	. . . . . . 7 ⊢ 𝑥 ∈ V
13	12	biantrur 530	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵)))
14	13	opabbii 5215	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵))}
15	11, 14	eqtri 2759	. . . 4 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵))}
16	5, 7, 10, 15	fvopab3ig 6994	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ 𝑅) → (𝜓 → (𝐹‘𝐴) = 𝐶))
17	1, 16	sylan 579	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅) → (𝜓 → (𝐹‘𝐴) = 𝐶))
18	17	3impia 1116	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅 ∧ 𝜓) → (𝐹‘𝐴) = 𝐶)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∃*wmo 2531  Vcvv 3473  {copab 5210  ‘cfv 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551

This theorem is referenced by: (None)