Theorem fvopabf4 4340

Description: Special case of fvopab4 3780 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 4334	. 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 4334	. . . . . 6 |- (x e. (R ^m D) <-> x:D-->R)
7	6	anbi1i 481	. . . . 5 |- ((x e. (R ^m D) /\ y = B) <-> (x:D-->R /\ y = B))
8	7	opabbii 2671	. . . 4 |- {<.x, y>. | (x e. (R ^m D) /\ y = B)} = {<.x, y>. | (x:D-->R /\ y = B)}
9	5, 8	eqtr4 1498	. . 3 |- F = {<.x, y>. | (x e. (R ^m D) /\ y = B)}
10		fvopabf4.1	. . 3 |- C e. V
11	4, 9, 10	fvopab4 3780	. 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 956   e. wcel 958  Vcvv 1811  {copab 2666  -->wf 3178  ` cfv 3182  (class class class)co 3963   ^m cm 4322

This theorem is referenced by:  nmoval 8426  nmopvalt 9782  nmfnvalt 9803  nlfnvalt 9808  eigvecvalt 9822  eigvalvalt 9823  specvalt 9824

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-oprab 3966  df-map 4324

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