Theorem mapvalg 6892

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 6888	. . 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 2825	. . 3 |- ( A e. C -> A e. _V )
4		elex 2825	. . 3 |- ( B e. D -> B e. _V )
5		feq3 5493	. . . . . 6 |- ( x = A -> ( f : y --> x <-> f : y --> A ) )
6	5	abbidv 2352	. . . . 5 |- ( x = A -> { f | f : y --> x } = { f | f : y --> A } )
7		feq2 5492	. . . . . 6 |- ( y = B -> ( f : y --> A <-> f : B --> A ) )
8	7	abbidv 2352	. . . . 5 |- ( y = B -> { f | f : y --> A } = { f | f : B --> A } )
9		df-map 6884	. . . . 5 |- ^m = ( x e. _V , y e. _V |-> { f | f : y --> x } )
10	6, 8, 9	ovmpog 6188	. . . 4 |- ( ( A e. _V /\ B e. _V /\ { f | f : B --> A } e. _V ) -> ( A ^m B ) = { f | f : B --> A } )
11	10	3expia 1232	. . 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 1398   e. wcel 2203   {cab 2218   _Vcvv 2813   -->wf 5348  (class class class)co 6050   ^m cmap 6882

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-map 6884

This theorem is referenced by:  mapval  6894  elmapg  6895  ixpconstg  6942  ptex  13477  psrval  14814  psrbasg  14829  cnovex  15061  ispsmet  15188  cncfval  15437