Theorem mapex 4312

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 3622	. . . 4 |- (f:A-->B -> f (_ (A X. B))
2	1	ss2abi 2110	. . 3 |- {f | f:A-->B} (_ {f | f (_ (A X. B)}
3		df-pw 2392	. . 3 |- P~(A X. B) = {f | f (_ (A X. B)}
4	2, 3	sseqtr4 2084	. 2 |- {f | f:A-->B} (_ P~(A X. B)
5		ssexg 2711	. . 3 |- (({f | f:A-->B} (_ P~(A X. B) /\ P~(A X. B) e. V) -> {f | f:A-->B} e. V)
6		xpexg 3249	. . . 4 |- ((A e. C /\ B e. D) -> (A X. B) e. V)
7		pwexg 2736	. . . 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 693	1 |- ((A e. C /\ B e. D) -> {f | f:A-->B} e. V)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 955  {cab 1456  Vcvv 1802   (_ wss 2037  P~cpw 2391   X. cxp 3158  -->wf 3168

This theorem is referenced by:  fnmap 4313  mapvalg 4314  cncfval 7199  infxpidmlem9 7503  homeofval 10403  isfuna 10592

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

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