Theorem mapex 4328

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 3637	. . . 4 |- (f:A-->B -> f (_ (A X. B))
2	1	ss2abi 2120	. . 3 |- {f | f:A-->B} (_ {f | f (_ (A X. B)}
3		df-pw 2402	. . 3 |- P~(A X. B) = {f | f (_ (A X. B)}
4	2, 3	sseqtr4 2094	. 2 |- {f | f:A-->B} (_ P~(A X. B)
5		ssexg 2721	. . 3 |- (({f | f:A-->B} (_ P~(A X. B) /\ P~(A X. B) e. V) -> {f | f:A-->B} e. V)
6		xpexg 3259	. . . 4 |- ((A e. C /\ B e. D) -> (A X. B) e. V)
7		pwexg 2746	. . . 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 451	. 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 695	1 |- ((A e. C /\ B e. D) -> {f | f:A-->B} e. V)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 958  {cab 1463  Vcvv 1811   (_ wss 2047  P~cpw 2401   X. cxp 3168  -->wf 3178

This theorem is referenced by:  fnmap 4329  mapvalg 4330  cncfval 7264  infxpidmlem9 7560  homeofval 10516  isfuna 10754

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

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