Theorem mapex 4334

Description: The class of all functions mapping one set to another is a set. Remark after Definition 10.24 of [Kunen] p. 31. (Contributed by Raph Levien, 4-Dec-2003.)

Assertion

Ref	Expression
mapex	|- ((A e. C /\ B e. D) -> {f | f:A-->B} e. V)

Distinct variable groups:   A,f   B,f

Proof of Theorem mapex

Step	Hyp	Ref	Expression
1		fssxp 3643	. . . 4 |- (f:A-->B -> f (_ (A X. B))
2	1	ss2abi 2123	. . 3 |- {f | f:A-->B} (_ {f | f (_ (A X. B)}
3		df-pw 2406	. . 3 |- P~(A X. B) = {f | f (_ (A X. B)}
4	2, 3	sseqtr4 2097	. 2 |- {f | f:A-->B} (_ P~(A X. B)
5		ssexg 2726	. . 3 |- (({f | f:A-->B} (_ P~(A X. B) /\ P~(A X. B) e. V) -> {f | f:A-->B} e. V)
6		xpexg 3265	. . . 4 |- ((A e. C /\ B e. D) -> (A X. B) e. V)
7		pwexg 2752	. . . 4 |- ((A X. B) e. V -> P~(A X. B) e. V)
8	6, 7	syl 10	. . 3 |- ((A e. C /\ B e. D) -> P~(A X. B) e. V)
9	5, 8	sylan2 453	. 2 |- (({f | f:A-->B} (_ P~(A X. B) /\ (A e. C /\ B e. D)) -> {f | f:A-->B} e. V)
10	4, 9	mpan 697	1 |- ((A e. C /\ B e. D) -> {f | f:A-->B} e. V)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 960  {cab 1466  Vcvv 1814   (_ wss 2050  P~cpw 2405   X. cxp 3174  -->wf 3184

This theorem is referenced by:  fnmap 4335  mapvalg 4336  cncfval 7264  infxpidmlem9 7561  homeofval 10502  isfuna 10725

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-cnv 3192  df-dm 3194  df-rn 3195  df-fun 3198  df-fn 3199  df-f 3200

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