Theorem fvopab4g 2870

Description: Value of a function given by ordered pair abstraction.

Hypotheses

Ref	Expression
fvopab4g.1	⊢ (x = A → B = C)
fvopab4g.2	⊢ F = {⟨x, y⟩∣(x ∈ D ∧ y = B)}

Assertion

Ref	Expression
fvopab4g	⊢ ((A ∈ D ∧ C ∈ R) → (F ‘A) = C)

Distinct variable group(s):   x,y,A   y,B   x,C,y   x,D,y

Proof of Theorem fvopab4g

Step	Hyp	Ref	Expression
1		cleqid 1102	. 2 ⊢ C = C
2		fvopab4g.1	. . . 4 ⊢ (x = A → B = C)
3	2	cleq2d 1112	. . 3 ⊢ (x = A → (y = B ↔ y = C))
4		cleq1 1107	. . 3 ⊢ (y = C → (y = C ↔ C = C))
5		moeq 1431	. . . 4 ⊢ ∃*y y = B
6	5	a1i 7	. . 3 ⊢ (x ∈ D → ∃*y y = B)
7		fvopab4g.2	. . 3 ⊢ F = {⟨x, y⟩∣(x ∈ D ∧ y = B)}
8	3, 4, 6, 7	fvopab3ig 2869	. 2 ⊢ ((A ∈ D ∧ C ∈ R) → (C = C → (F ‘A) = C))
9	1, 8	mpi 44	1 ⊢ ((A ∈ D ∧ C ∈ R) → (F ‘A) = C)

Syntax hints:   → wi 2   ∧ wa 196  ∃*wmo 1008   = wceq 1091   ∈ wcel 1092  {copab 2055   ‘cfv 2422

This theorem is referenced by:  fvopab4 2871  fvopabg 2872  cfval 3701  pjvalt 5246  spanvalt 5300  hsupval2t 5301

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438

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