Theorem fvopabf4 4346

Description: Special case of fvopab4 3786 for operator theorems.

Hypotheses

Ref	Expression
fvopabf4.1	|- C e. V
fvopabf4.2	|- D e. V
fvopabf4.3	|- R e. V
fvopabf4.4	|- (x = A -> B = C)
fvopabf4.5	|- F = {<.x, y>. | (x:D-->R /\ y = B)}

Assertion

Ref	Expression
fvopabf4	|- (A:D-->R -> (F` A) = C)

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

Proof of Theorem fvopabf4

Step	Hyp	Ref	Expression
1		fvopabf4.3	. . 3 |- R e. V
2		fvopabf4.2	. . 3 |- D e. V
3	1, 2	elmap 4340	. 2 |- (A e. (R ^m D) <-> A:D-->R)
4		fvopabf4.4	. . 3 |- (x = A -> B = C)
5		fvopabf4.5	. . . 4 |- F = {<.x, y>. | (x:D-->R /\ y = B)}
6	1, 2	elmap 4340	. . . . . 6 |- (x e. (R ^m D) <-> x:D-->R)
7	6	anbi1i 483	. . . . 5 |- ((x e. (R ^m D) /\ y = B) <-> (x:D-->R /\ y = B))
8	7	opabbii 2676	. . . 4 |- {<.x, y>. | (x e. (R ^m D) /\ y = B)} = {<.x, y>. | (x:D-->R /\ y = B)}
9	5, 8	eqtr4 1501	. . 3 |- F = {<.x, y>. | (x e. (R ^m D) /\ y = B)}
10		fvopabf4.1	. . 3 |- C e. V
11	4, 9, 10	fvopab4 3786	. 2 |- (A e. (R ^m D) -> (F` A) = C)
12	3, 11	sylbir 201	1 |- (A:D-->R -> (F` A) = C)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  Vcvv 1814  {copab 2671  -->wf 3184  ` cfv 3188  (class class class)co 3969   ^m cm 4328

This theorem is referenced by:  nmoval 8422  nmopvalt 9777  nmfnvalt 9798  nlfnvalt 9803  eigvecvalt 9817  eigvalvalt 9818  specvalt 9819

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-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  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-fn 3199  df-f 3200  df-fv 3204  df-opr 3971  df-oprab 3972  df-map 4330

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