Theorem oprabvali 4025

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 4024	. 2 |- ((A e. R /\ B e. S /\ C e. V) -> (th -> (AFB) = C))
8	1, 7	mp3an3 905	1 |- ((A e. R /\ B e. S) -> (th -> (AFB) = C))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E*wmo 1381  Vcvv 1811  (class class class)co 3963  {copab2 3964

This theorem is referenced by:  oprabval3 4030  th3q 4317

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-sep 2703  ax-pow 2742  ax-pr 2779

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-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-fv 3198  df-opr 3965  df-oprab 3966

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