Theorem oprabex2g 4020

Description: Existence of an operation class abstraction (special case).

Hypothesis

Ref	Expression
oprabex2g.1	|- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}

Assertion

Ref	Expression
oprabex2g	|- ((A e. R /\ B e. S) -> F e. V)

Distinct variable groups:   x,y,z,A   x,B,y,z   z,C

Proof of Theorem oprabex2g

Step	Hyp	Ref	Expression
1		ssexg 2721	. . . 4 |- ((dom {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} (_ (A X. B) /\ (A X. B) e. V) -> dom {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} e. V)
2		dmoprabss 4003	. . . . 5 |- dom {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} (_ (A X. B)
3	2	a1i 8	. . . 4 |- ((A e. R /\ B e. S) -> dom {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} (_ (A X. B))
4		xpexg 3259	. . . 4 |- ((A e. R /\ B e. S) -> (A X. B) e. V)
5	1, 3, 4	sylanc 471	. . 3 |- ((A e. R /\ B e. S) -> dom {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} e. V)
6		moeq 1920	. . . . . 6 |- E*z z = C
7	6	moani 1423	. . . . 5 |- E*z((x e. A /\ y e. B) /\ z = C)
8	7	funoprab 4011	. . . 4 |- Fun {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}
9		funex 3608	. . . 4 |- ((Fun {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} /\ dom {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} e. V) -> {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} e. V)
10	8, 9	mpan 695	. . 3 |- (dom {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} e. V -> {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} e. V)
11	5, 10	syl 10	. 2 |- ((A e. R /\ B e. S) -> {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} e. V)
12		oprabex2g.1	. 2 |- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}
13	11, 12	syl5eqel 1552	1 |- ((A e. R /\ B e. S) -> F e. V)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811   (_ wss 2047   X. cxp 3168  dom cdm 3170  Fun wfun 3176  {copab2 3964

This theorem is referenced by:  oprabex2 4021  blfval 7835  grpdivfval 8081  hmeogrp 10538

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-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-fn 3193  df-oprab 3966

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