Theorem mapsnen 6879

Description: Set exponentiation to a singleton exponent is equinumerous to its base. Exercise 4.43 of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.) (Revised by Mario Carneiro, 15-Nov-2014.)

Hypotheses

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

Assertion

Ref	Expression
mapsnen	|- ( A ^m { B } ) ~~ A

Proof of Theorem mapsnen

Dummy variables y w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnmap 6723	. . 3 |- ^m Fn ( _V X. _V )
2		mapsnen.1	. . 3 |- A e. _V
3		mapsnen.2	. . . 4 |- B e. _V
4	3	snex 4219	. . 3 |- { B } e. _V
5		fnovex 5958	. . 3 |- ( ( ^m Fn ( _V X. _V ) /\ A e. _V /\ { B } e. _V ) -> ( A ^m { B } ) e. _V )
6	1, 2, 4, 5	mp3an 1348	. 2 |- ( A ^m { B } ) e. _V
7		vex 2766	. . . 4 |- z e. _V
8	7, 3	fvex 5581	. . 3 |- ( z ` B ) e. _V
9	8	a1i 9	. 2 |- ( z e. ( A ^m { B } ) -> ( z ` B ) e. _V )
10		vex 2766	. . . . 5 |- w e. _V
11	3, 10	opex 4263	. . . 4 |- <. B , w >. e. _V
12	11	snex 4219	. . 3 |- { <. B , w >. } e. _V
13	12	a1i 9	. 2 |- ( w e. A -> { <. B , w >. } e. _V )
14	2, 3	mapsn 6758	. . . . . 6 |- ( A ^m { B } ) = { z | E. y e. A z = { <. B , y >. } }
15	14	abeq2i 2307	. . . . 5 |- ( z e. ( A ^m { B } ) <-> E. y e. A z = { <. B , y >. } )
16	15	anbi1i 458	. . . 4 |- ( ( z e. ( A ^m { B } ) /\ w = ( z ` B ) ) <-> ( E. y e. A z = { <. B , y >. } /\ w = ( z ` B ) ) )
17		r19.41v 2653	. . . 4 |- ( E. y e. A ( z = { <. B , y >. } /\ w = ( z ` B ) ) <-> ( E. y e. A z = { <. B , y >. } /\ w = ( z ` B ) ) )
18		df-rex 2481	. . . 4 |- ( E. y e. A ( z = { <. B , y >. } /\ w = ( z ` B ) ) <-> E. y ( y e. A /\ ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) )
19	16, 17, 18	3bitr2i 208	. . 3 |- ( ( z e. ( A ^m { B } ) /\ w = ( z ` B ) ) <-> E. y ( y e. A /\ ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) )
20		fveq1 5560	. . . . . . . . . 10 |- ( z = { <. B , y >. } -> ( z ` B ) = ( { <. B , y >. } ` B ) )
21		vex 2766	. . . . . . . . . . 11 |- y e. _V
22	3, 21	fvsn 5760	. . . . . . . . . 10 |- ( { <. B , y >. } ` B ) = y
23	20, 22	eqtrdi 2245	. . . . . . . . 9 |- ( z = { <. B , y >. } -> ( z ` B ) = y )
24	23	eqeq2d 2208	. . . . . . . 8 |- ( z = { <. B , y >. } -> ( w = ( z ` B ) <-> w = y ) )
25		equcom 1720	. . . . . . . 8 |- ( w = y <-> y = w )
26	24, 25	bitrdi 196	. . . . . . 7 |- ( z = { <. B , y >. } -> ( w = ( z ` B ) <-> y = w ) )
27	26	pm5.32i 454	. . . . . 6 |- ( ( z = { <. B , y >. } /\ w = ( z ` B ) ) <-> ( z = { <. B , y >. } /\ y = w ) )
28	27	anbi2i 457	. . . . 5 |- ( ( y e. A /\ ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) <-> ( y e. A /\ ( z = { <. B , y >. } /\ y = w ) ) )
29		anass 401	. . . . 5 |- ( ( ( y e. A /\ z = { <. B , y >. } ) /\ y = w ) <-> ( y e. A /\ ( z = { <. B , y >. } /\ y = w ) ) )
30		ancom 266	. . . . 5 |- ( ( ( y e. A /\ z = { <. B , y >. } ) /\ y = w ) <-> ( y = w /\ ( y e. A /\ z = { <. B , y >. } ) ) )
31	28, 29, 30	3bitr2i 208	. . . 4 |- ( ( y e. A /\ ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) <-> ( y = w /\ ( y e. A /\ z = { <. B , y >. } ) ) )
32	31	exbii 1619	. . 3 |- ( E. y ( y e. A /\ ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) <-> E. y ( y = w /\ ( y e. A /\ z = { <. B , y >. } ) ) )
33		eleq1w 2257	. . . . 5 |- ( y = w -> ( y e. A <-> w e. A ) )
34		opeq2 3810	. . . . . . 7 |- ( y = w -> <. B , y >. = <. B , w >. )
35	34	sneqd 3636	. . . . . 6 |- ( y = w -> { <. B , y >. } = { <. B , w >. } )
36	35	eqeq2d 2208	. . . . 5 |- ( y = w -> ( z = { <. B , y >. } <-> z = { <. B , w >. } ) )
37	33, 36	anbi12d 473	. . . 4 |- ( y = w -> ( ( y e. A /\ z = { <. B , y >. } ) <-> ( w e. A /\ z = { <. B , w >. } ) ) )
38	10, 37	ceqsexv 2802	. . 3 |- ( E. y ( y = w /\ ( y e. A /\ z = { <. B , y >. } ) ) <-> ( w e. A /\ z = { <. B , w >. } ) )
39	19, 32, 38	3bitri 206	. 2 |- ( ( z e. ( A ^m { B } ) /\ w = ( z ` B ) ) <-> ( w e. A /\ z = { <. B , w >. } ) )
40	6, 2, 9, 13, 39	en2i 6838	1 |- ( A ^m { B } ) ~~ A

Syntax hints:   /\ wa 104   = wceq 1364   E.wex 1506   e. wcel 2167   E.wrex 2476   _Vcvv 2763   {csn 3623   <.cop 3626   class class class wbr 4034   X. cxp 4662   Fn wfn 5254   ` cfv 5259  (class class class)co 5925   ^m cmap 6716   ~~ cen 6806

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-map 6718  df-en 6809

This theorem is referenced by:  exmidpw2en  6982