Theorem mapvalg 6714

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 6710	. . 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 2771	. . 3 |- ( A e. C -> A e. _V )
4		elex 2771	. . 3 |- ( B e. D -> B e. _V )
5		feq3 5389	. . . . . 6 |- ( x = A -> ( f : y --> x <-> f : y --> A ) )
6	5	abbidv 2311	. . . . 5 |- ( x = A -> { f | f : y --> x } = { f | f : y --> A } )
7		feq2 5388	. . . . . 6 |- ( y = B -> ( f : y --> A <-> f : B --> A ) )
8	7	abbidv 2311	. . . . 5 |- ( y = B -> { f | f : y --> A } = { f | f : B --> A } )
9		df-map 6706	. . . . 5 |- ^m = ( x e. _V , y e. _V |-> { f | f : y --> x } )
10	6, 8, 9	ovmpog 6054	. . . 4 |- ( ( A e. _V /\ B e. _V /\ { f | f : B --> A } e. _V ) -> ( A ^m B ) = { f | f : B --> A } )
11	10	3expia 1207	. . 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 1364   e. wcel 2164   {cab 2179   _Vcvv 2760   -->wf 5251  (class class class)co 5919   ^m cmap 6704

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-map 6706

This theorem is referenced by:  mapval  6716  elmapg  6717  ixpconstg  6763  ptex  12878  psrval  14163  psrbasg  14170  cnovex  14375  ispsmet  14502  cncfval  14751