Theorem f1oabexg 3685

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 1468	. . . 4 |- {f | (f:A-->B /\ ph)} = {f | (f:A-->B /\ ph)}
2	1	fabexg 3638	. . 3 |- ((A e. C /\ B e. D) -> {f | (f:A-->B /\ ph)} e. V)
3		f1of 3674	. . . . . 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 2110	. . . 4 |- {f | (f:A-1-1-onto->B /\ ph)} (_ {f | (f:A-->B /\ ph)}
6		ssexg 2711	. . . 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 693	. . 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 1544	1 |- ((A e. C /\ B e. D) -> F e. V)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  {cab 1456  Vcvv 1802   (_ wss 2037  -->wf 3168  -1-1-onto->wf1o 3171

This theorem is referenced by:  symgval 10308  symggrpiOLD 10311  symggrpi 10313

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-xp 3174  df-rel 3175  df-cnv 3176  df-dm 3178  df-rn 3179  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-f1o 3187

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