Theorem oprabex2g 4015

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 2717	. . . 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 3998	. . . . 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 3255	. . . 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 1917	. . . . . 6 |- E*z z = C
7	6	moani 1422	. . . . 5 |- E*z((x e. A /\ y e. B) /\ z = C)
8	7	funoprab 4006	. . . 4 |- Fun {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}
9		funex 3604	. . . 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 694	. . 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 1550	1 |- ((A e. R /\ B e. S) -> F e. V)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957  Vcvv 1808   (_ wss 2044   X. cxp 3164  dom cdm 3166  Fun wfun 3172  {copab2 3959

This theorem is referenced by:  oprabex2 4016  blfval 7797  grpdivfval 8043  hmeogrp 10484

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-oprab 3961

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