Theorem fvopab4s 3789

Description: Value of a function given by ordered-pair class abstraction, using explicit class substitution.

Hypotheses

Ref	Expression
fvopab4s.1	|- A e. V
fvopab4s.2	|- B e. V
fvopab4s.3	|- F = {<.x, y>. | (x e. C /\ y = B)}

Assertion

Ref	Expression
fvopab4s	|- (A e. C -> (F` A) = [_A / x]_B)

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

Proof of Theorem fvopab4s

Step	Hyp	Ref	Expression
1		fvopab4s.1	. 2 |- A e. V
2		fvopab4s.2	. 2 |- B e. V
3		ax-17 973	. 2 |- (z e. A -> A.x z e. A)
4		fvopab4s.3	. 2 |- F = {<.x, y>. | (x e. C /\ y = B)}
5	1, 2, 3, 4	fvopab4sf 3788	1 |- (A e. C -> (F` A) = [_A / x]_B)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  Vcvv 1814  [_csb 2004  {copab 2671  ` cfv 3188

This theorem is referenced by:  fopabcos 3839

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204

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