Theorem mapvalg 6805

Description: The value of set exponentiation. ( A ^m B ) is the set of all functions that map from B to A. Definition 10.24 of [Kunen] p. 24. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
mapvalg	|- ( ( A e. C /\ B e. D ) -> ( A ^m B ) = { f | f : B --> A } )

Distinct variable groups:   A, f   B, f

Allowed substitution hints:   C( f)   D( f)

Proof of Theorem mapvalg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapex 6801	. . 3 |- ( ( B e. D /\ A e. C ) -> { f | f : B --> A } e. _V )
2	1	ancoms 268	. 2 |- ( ( A e. C /\ B e. D ) -> { f | f : B --> A } e. _V )
3		elex 2811	. . 3 |- ( A e. C -> A e. _V )
4		elex 2811	. . 3 |- ( B e. D -> B e. _V )
5		feq3 5458	. . . . . 6 |- ( x = A -> ( f : y --> x <-> f : y --> A ) )
6	5	abbidv 2347	. . . . 5 |- ( x = A -> { f | f : y --> x } = { f | f : y --> A } )
7		feq2 5457	. . . . . 6 |- ( y = B -> ( f : y --> A <-> f : B --> A ) )
8	7	abbidv 2347	. . . . 5 |- ( y = B -> { f | f : y --> A } = { f | f : B --> A } )
9		df-map 6797	. . . . 5 |- ^m = ( x e. _V , y e. _V |-> { f | f : y --> x } )
10	6, 8, 9	ovmpog 6139	. . . 4 |- ( ( A e. _V /\ B e. _V /\ { f | f : B --> A } e. _V ) -> ( A ^m B ) = { f | f : B --> A } )
11	10	3expia 1229	. . 3 |- ( ( A e. _V /\ B e. _V ) -> ( { f | f : B --> A } e. _V -> ( A ^m B ) = { f | f : B --> A } ) )
12	3, 4, 11	syl2an 289	. 2 |- ( ( A e. C /\ B e. D ) -> ( { f | f : B --> A } e. _V -> ( A ^m B ) = { f | f : B --> A } ) )
13	2, 12	mpd 13	1 |- ( ( A e. C /\ B e. D ) -> ( A ^m B ) = { f | f : B --> A } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   {cab 2215   _Vcvv 2799   -->wf 5314  (class class class)co 6001   ^m cmap 6795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-map 6797

This theorem is referenced by:  mapval  6807  elmapg  6808  ixpconstg  6854  ptex  13297  psrval  14630  psrbasg  14638  cnovex  14870  ispsmet  14997  cncfval  15246