Theorem mapval2 6825

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 5779	. . . 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 6824	. . 3 |- ( g e. ( A ^m B ) <-> g : B --> A )
7		elin 3387	. . . 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 3656	. . . . 5 |- ( g e. ~P ( B X. A ) <-> g C_ ( B X. A ) )
9		vex 2802	. . . . . 6 |- g e. _V
10		fneq1 5409	. . . . . 6 |- ( f = g -> ( f Fn B <-> g Fn B ) )
11	9, 10	elab 2947	. . . . 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 2226	1 |- ( A ^m B ) = ( ~P ( B X. A ) i^i { f | f Fn B } )

Syntax hints:   /\ wa 104   = wceq 1395   e. wcel 2200   {cab 2215   _Vcvv 2799   i^i cin 3196   C_ wss 3197   ~Pcpw 3649   X. cxp 4717   Fn wfn 5313   -->wf 5314  (class class class)co 6001   ^m cmap 6795

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-map 6797

This theorem is referenced by: (None)