Theorem mapval2 6565

Description: Alternate expression for the value of set exponentiation. (Contributed by NM, 3-Nov-2007.)

Hypotheses

Ref	Expression
elmap.1	|- A e. _V
elmap.2	|- B e. _V

Assertion

Ref	Expression
mapval2	|- ( A ^m B ) = ( ~P ( B X. A ) i^i { f | f Fn B } )

Distinct variable group:   B, f

Allowed substitution hint:   A( f)

Proof of Theorem mapval2

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dff2 5557	. . . 4 |- ( g : B --> A <-> ( g Fn B /\ g C_ ( B X. A ) ) )
2		ancom 264	. . . 4 |- ( ( g Fn B /\ g C_ ( B X. A ) ) <-> ( g C_ ( B X. A ) /\ g Fn B ) )
3	1, 2	bitri 183	. . 3 |- ( g : B --> A <-> ( g C_ ( B X. A ) /\ g Fn B ) )
4		elmap.1	. . . 4 |- A e. _V
5		elmap.2	. . . 4 |- B e. _V
6	4, 5	elmap 6564	. . 3 |- ( g e. ( A ^m B ) <-> g : B --> A )
7		elin 3254	. . . 4 |- ( g e. ( ~P ( B X. A ) i^i { f | f Fn B } ) <-> ( g e. ~P ( B X. A ) /\ g e. { f | f Fn B } ) )
8		velpw 3512	. . . . 5 |- ( g e. ~P ( B X. A ) <-> g C_ ( B X. A ) )
9		vex 2684	. . . . . 6 |- g e. _V
10		fneq1 5206	. . . . . 6 |- ( f = g -> ( f Fn B <-> g Fn B ) )
11	9, 10	elab 2823	. . . . 5 |- ( g e. { f | f Fn B } <-> g Fn B )
12	8, 11	anbi12i 455	. . . 4 |- ( ( g e. ~P ( B X. A ) /\ g e. { f | f Fn B } ) <-> ( g C_ ( B X. A ) /\ g Fn B ) )
13	7, 12	bitri 183	. . 3 |- ( g e. ( ~P ( B X. A ) i^i { f | f Fn B } ) <-> ( g C_ ( B X. A ) /\ g Fn B ) )
14	3, 6, 13	3bitr4i 211	. 2 |- ( g e. ( A ^m B ) <-> g e. ( ~P ( B X. A ) i^i { f | f Fn B } ) )
15	14	eqriv 2134	1 |- ( A ^m B ) = ( ~P ( B X. A ) i^i { f | f Fn B } )

Syntax hints:   /\ wa 103   = wceq 1331   e. wcel 1480   {cab 2123   _Vcvv 2681   i^i cin 3065   C_ wss 3066   ~Pcpw 3505   X. cxp 4532   Fn wfn 5113   -->wf 5114  (class class class)co 5767   ^m cmap 6535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-map 6537

This theorem is referenced by: (None)