Theorem oprabvali 3049

Description: The value of an operation 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 group(s):   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 3048	. 2 |- ((A e. R /\ B e. S /\ C e. V) -> (th -> (AFB) = C))
8	1, 7	mp3an3 641	1 |- ((A e. R /\ B e. S) -> (th -> (AFB) = C))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E*wmo 1008   = wceq 1091   e. wcel 1092  Vcvv 1348  (class class class)co 3001  {copab2 3002

This theorem is referenced by:  oprabval3 3052  th3q 3253

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-opr 3003  df-oprab 3004

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