Theorem mapval2 6925

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 5826	. . . 4 |- ( g : B --> A <-> ( g Fn B /\ g C_ ( B X. A ) ) )
2		ancom 266	. . . 4 |- ( ( g Fn B /\ g C_ ( B X. A ) ) <-> ( g C_ ( B X. A ) /\ g Fn B ) )
3	1, 2	bitri 184	. . 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 6924	. . 3 |- ( g e. ( A ^m B ) <-> g : B --> A )
7		elin 3406	. . . 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 3681	. . . . 5 |- ( g e. ~P ( B X. A ) <-> g C_ ( B X. A ) )
9		vex 2818	. . . . . 6 |- g e. _V
10		fneq1 5449	. . . . . 6 |- ( f = g -> ( f Fn B <-> g Fn B ) )
11	9, 10	elab 2964	. . . . 5 |- ( g e. { f | f Fn B } <-> g Fn B )
12	8, 11	anbi12i 460	. . . 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 184	. . 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 212	. 2 |- ( g e. ( A ^m B ) <-> g e. ( ~P ( B X. A ) i^i { f | f Fn B } ) )
15	14	eqriv 2231	1 |- ( A ^m B ) = ( ~P ( B X. A ) i^i { f | f Fn B } )

Syntax hints:   /\ wa 104   = wceq 1398   e. wcel 2205   {cab 2220   _Vcvv 2815   i^i cin 3213   C_ wss 3214   ~Pcpw 3674   X. cxp 4752   Fn wfn 5352   -->wf 5353  (class class class)co 6058   ^m cmap 6895

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-map 6897

This theorem is referenced by: (None)