Theorem mapvalg 6726

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 6722	. . 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 2774	. . 3 |- ( A e. C -> A e. _V )
4		elex 2774	. . 3 |- ( B e. D -> B e. _V )
5		feq3 5395	. . . . . 6 |- ( x = A -> ( f : y --> x <-> f : y --> A ) )
6	5	abbidv 2314	. . . . 5 |- ( x = A -> { f | f : y --> x } = { f | f : y --> A } )
7		feq2 5394	. . . . . 6 |- ( y = B -> ( f : y --> A <-> f : B --> A ) )
8	7	abbidv 2314	. . . . 5 |- ( y = B -> { f | f : y --> A } = { f | f : B --> A } )
9		df-map 6718	. . . . 5 |- ^m = ( x e. _V , y e. _V |-> { f | f : y --> x } )
10	6, 8, 9	ovmpog 6061	. . . 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 2167   {cab 2182   _Vcvv 2763   -->wf 5255  (class class class)co 5925   ^m cmap 6716

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-map 6718

This theorem is referenced by:  mapval  6728  elmapg  6729  ixpconstg  6775  ptex  12966  psrval  14296  psrbasg  14303  cnovex  14516  ispsmet  14643  cncfval  14892