Theorem oprabvali 4031

Description: The value of an operation class abstraction (weak version).

Hypotheses

Ref	Expression
oprabvali.1	|- C e. V
oprabvali.2	|- (x = A -> (ph <-> ps))
oprabvali.3	|- (y = B -> (ps <-> ch))
oprabvali.4	|- (z = C -> (ch <-> th))
oprabvali.5	|- ((x e. R /\ y e. S) -> E*zph)
oprabvali.6	|- F = {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}

Assertion

Ref	Expression
oprabvali	|- ((A e. R /\ B e. S) -> (th -> (AFB) = C))

Distinct variable groups:   x,y,z,A   x,B,y,z   x,C,y,z   x,R,y,z   x,S,y,z   ps,x   ch,x,y   th,x,y,z

Proof of Theorem oprabvali

Step	Hyp	Ref	Expression
1		oprabvali.1	. 2 |- C e. V
2		oprabvali.2	. . 3 |- (x = A -> (ph <-> ps))
3		oprabvali.3	. . 3 |- (y = B -> (ps <-> ch))
4		oprabvali.4	. . 3 |- (z = C -> (ch <-> th))
5		oprabvali.5	. . 3 |- ((x e. R /\ y e. S) -> E*zph)
6		oprabvali.6	. . 3 |- F = {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}
7	2, 3, 4, 5, 6	oprabvalig 4030	. 2 |- ((A e. R /\ B e. S /\ C e. V) -> (th -> (AFB) = C))
8	1, 7	mp3an3 907	1 |- ((A e. R /\ B e. S) -> (th -> (AFB) = C))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E*wmo 1383  Vcvv 1814  (class class class)co 3969  {copab2 3970

This theorem is referenced by:  oprabval3 4036  th3q 4323

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-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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