Theorem fvopab6 5654

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 2771	. . 3 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ V)
2		fvopab6.2	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3		fvopab6.3	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
4	3	eqeq2d 2205	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐶))
5	2, 4	anbi12d 473	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝑦 = 𝐵) ↔ (𝜓 ∧ 𝑦 = 𝐶)))
6		iba 300	. . . . 5 ⊢ (𝑦 = 𝐶 → (𝜓 ↔ (𝜓 ∧ 𝑦 = 𝐶)))
7	6	bicomd 141	. . . 4 ⊢ (𝑦 = 𝐶 → ((𝜓 ∧ 𝑦 = 𝐶) ↔ 𝜓))
8		moeq 2935	. . . . . 6 ⊢ ∃*𝑦 𝑦 = 𝐵
9	8	moani 2112	. . . . 5 ⊢ ∃*𝑦(𝜑 ∧ 𝑦 = 𝐵)
10	9	a1i 9	. . . 4 ⊢ (𝑥 ∈ V → ∃*𝑦(𝜑 ∧ 𝑦 = 𝐵))
11		fvopab6.1	. . . . 5 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐵)}
12		vex 2763	. . . . . . 7 ⊢ 𝑥 ∈ V
13	12	biantrur 303	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵)))
14	13	opabbii 4096	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵))}
15	11, 14	eqtri 2214	. . . 4 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ (𝜑 ∧ 𝑦 = 𝐵))}
16	5, 7, 10, 15	fvopab3ig 5631	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ 𝑅) → (𝜓 → (𝐹‘𝐴) = 𝐶))
17	1, 16	sylan 283	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅) → (𝜓 → (𝐹‘𝐴) = 𝐶))
18	17	3impia 1202	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅 ∧ 𝜓) → (𝐹‘𝐴) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364  ∃*wmo 2043   ∈ wcel 2164  Vcvv 2760  {copab 4089  ‘cfv 5254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262

This theorem is referenced by: (None)