Theorem oprabval5 4035

Description: The value of an operation class abstraction. Special case.

Hypotheses

Ref	Expression
oprabval5.1	|- S e. V
oprabval5.2	|- (x = A -> R = G)
oprabval5.3	|- (y = B -> G = S)
oprabval5.4	|- F = {<.<.x, y>., z>. | z = R}

Assertion

Ref	Expression
oprabval5	|- ((A e. C /\ B e. D) -> (AFB) = S)

Distinct variable groups:   x,y,z,A   x,B,y,z   x,G   z,R   y,S,z

Proof of Theorem oprabval5

Step	Hyp	Ref	Expression
1		oprabval5.1	. . 3 |- S e. V
2		oprabval5.2	. . 3 |- (x = A -> R = G)
3		oprabval5.3	. . 3 |- (y = B -> G = S)
4		oprabval5.4	. . . 4 |- F = {<.<.x, y>., z>. | z = R}
5		visset 1816	. . . . . . 7 |- x e. V
6		visset 1816	. . . . . . 7 |- y e. V
7	5, 6	pm3.2i 285	. . . . . 6 |- (x e. V /\ y e. V)
8	7	biantrur 727	. . . . 5 |- (z = R <-> ((x e. V /\ y e. V) /\ z = R))
9	8	oprabbii 4003	. . . 4 |- {<.<.x, y>., z>. | z = R} = {<.<.x, y>., z>. | ((x e. V /\ y e. V) /\ z = R)}
10	4, 9	eqtr 1498	. . 3 |- F = {<.<.x, y>., z>. | ((x e. V /\ y e. V) /\ z = R)}
11	1, 2, 3, 10	oprabval2 4034	. 2 |- ((A e. V /\ B e. V) -> (AFB) = S)
12		elisset 1820	. 2 |- (A e. C -> A e. V)
13		elisset 1820	. 2 |- (B e. D -> B e. V)
14	11, 12, 13	syl2an 456	1 |- ((A e. C /\ B e. D) -> (AFB) = S)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  Vcvv 1814  (class class class)co 3969  {copab2 3970

This theorem is referenced by:  1st2val 4101  2nd2val 4102  seq1val 6313  shftfval 6343  seq0fval 6536  seqzfval 6538  dfseq0 6564

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-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-fv 3204  df-opr 3971  df-oprab 3972

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