Theorem f1oabexg 3700

Description: The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.)

Hypothesis

Ref	Expression
f1oabexg.1	|- F = {f | (f:A-1-1-onto->B /\ ph)}

Assertion

Ref	Expression
f1oabexg	|- ((A e. C /\ B e. D) -> F e. V)

Distinct variable groups:   A,f   B,f

Proof of Theorem f1oabexg

Step	Hyp	Ref	Expression
1		eqid 1475	. . . 4 |- {f | (f:A-->B /\ ph)} = {f | (f:A-->B /\ ph)}
2	1	fabexg 3653	. . 3 |- ((A e. C /\ B e. D) -> {f | (f:A-->B /\ ph)} e. V)
3		f1of 3689	. . . . . 6 |- (f:A-1-1-onto->B -> f:A-->B)
4	3	anim1i 334	. . . . 5 |- ((f:A-1-1-onto->B /\ ph) -> (f:A-->B /\ ph))
5	4	ss2abi 2120	. . . 4 |- {f | (f:A-1-1-onto->B /\ ph)} (_ {f | (f:A-->B /\ ph)}
6		ssexg 2721	. . . 4 |- (({f | (f:A-1-1-onto->B /\ ph)} (_ {f | (f:A-->B /\ ph)} /\ {f | (f:A-->B /\ ph)} e. V) -> {f | (f:A-1-1-onto->B /\ ph)} e. V)
7	5, 6	mpan 695	. . 3 |- ({f | (f:A-->B /\ ph)} e. V -> {f | (f:A-1-1-onto->B /\ ph)} e. V)
8	2, 7	syl 10	. 2 |- ((A e. C /\ B e. D) -> {f | (f:A-1-1-onto->B /\ ph)} e. V)
9		f1oabexg.1	. 2 |- F = {f | (f:A-1-1-onto->B /\ ph)}
10	8, 9	syl5eqel 1552	1 |- ((A e. C /\ B e. D) -> F e. V)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  Vcvv 1811   (_ wss 2047  -->wf 3178  -1-1-onto->wf1o 3181

This theorem is referenced by:  symgval 10403  symggrpi 10406

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-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  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-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-xp 3184  df-rel 3185  df-cnv 3186  df-dm 3188  df-rn 3189  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-f1o 3197

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