Theorem mapval2 6952

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 5846	. . . 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 6951	. . 3 |- ( g e. ( A ^m B ) <-> g : B --> A )
7		elin 3412	. . . 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 3695	. . . . 5 |- ( g e. ~P ( B X. A ) <-> g C_ ( B X. A ) )
9		vex 2824	. . . . . 6 |- g e. _V
10		fneq1 5467	. . . . . 6 |- ( f = g -> ( f Fn B <-> g Fn B ) )
11	9, 10	elab 2970	. . . . 5 |- ( g e. { f | f Fn B } <-> g Fn B )
12	8, 11	anbi12i 464	. . . 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 2235	1 |- ( A ^m B ) = ( ~P ( B X. A ) i^i { f | f Fn B } )

Syntax hints:   /\ wa 104   = wceq 1402   e. wcel 2209   {cab 2224   _Vcvv 2821   i^i cin 3219   C_ wss 3220   ~Pcpw 3688   X. cxp 4770   Fn wfn 5370   -->wf 5371  (class class class)co 6078   ^m cmap 6915

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fv 5383  df-ov 6081  df-oprab 6082  df-mpo 6083  df-map 6917

This theorem is referenced by: (None)