Theorem mapvalg 6826

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 6822	. . 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 2814	. . 3 |- ( A e. C -> A e. _V )
4		elex 2814	. . 3 |- ( B e. D -> B e. _V )
5		feq3 5467	. . . . . 6 |- ( x = A -> ( f : y --> x <-> f : y --> A ) )
6	5	abbidv 2349	. . . . 5 |- ( x = A -> { f | f : y --> x } = { f | f : y --> A } )
7		feq2 5466	. . . . . 6 |- ( y = B -> ( f : y --> A <-> f : B --> A ) )
8	7	abbidv 2349	. . . . 5 |- ( y = B -> { f | f : y --> A } = { f | f : B --> A } )
9		df-map 6818	. . . . 5 |- ^m = ( x e. _V , y e. _V |-> { f | f : y --> x } )
10	6, 8, 9	ovmpog 6155	. . . 4 |- ( ( A e. _V /\ B e. _V /\ { f | f : B --> A } e. _V ) -> ( A ^m B ) = { f | f : B --> A } )
11	10	3expia 1231	. . 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 1397   e. wcel 2202   {cab 2217   _Vcvv 2802   -->wf 5322  (class class class)co 6017   ^m cmap 6816

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-map 6818

This theorem is referenced by:  mapval  6828  elmapg  6829  ixpconstg  6875  ptex  13346  psrval  14679  psrbasg  14687  cnovex  14919  ispsmet  15046  cncfval  15295