Theorem mapsnen 6805

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 6649	. . 3 |- ^m Fn ( _V X. _V )
2		mapsnen.1	. . 3 |- A e. _V
3		mapsnen.2	. . . 4 |- B e. _V
4	3	snex 4182	. . 3 |- { B } e. _V
5		fnovex 5902	. . 3 |- ( ( ^m Fn ( _V X. _V ) /\ A e. _V /\ { B } e. _V ) -> ( A ^m { B } ) e. _V )
6	1, 2, 4, 5	mp3an 1337	. 2 |- ( A ^m { B } ) e. _V
7		vex 2740	. . . 4 |- z e. _V
8	7, 3	fvex 5531	. . 3 |- ( z ` B ) e. _V
9	8	a1i 9	. 2 |- ( z e. ( A ^m { B } ) -> ( z ` B ) e. _V )
10		vex 2740	. . . . 5 |- w e. _V
11	3, 10	opex 4226	. . . 4 |- <. B , w >. e. _V
12	11	snex 4182	. . 3 |- { <. B , w >. } e. _V
13	12	a1i 9	. 2 |- ( w e. A -> { <. B , w >. } e. _V )
14	2, 3	mapsn 6684	. . . . . 6 |- ( A ^m { B } ) = { z | E. y e. A z = { <. B , y >. } }
15	14	abeq2i 2288	. . . . 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 2633	. . . 4 |- ( E. y e. A ( z = { <. B , y >. } /\ w = ( z ` B ) ) <-> ( E. y e. A z = { <. B , y >. } /\ w = ( z ` B ) ) )
18		df-rex 2461	. . . 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 5510	. . . . . . . . . 10 |- ( z = { <. B , y >. } -> ( z ` B ) = ( { <. B , y >. } ` B ) )
21		vex 2740	. . . . . . . . . . 11 |- y e. _V
22	3, 21	fvsn 5707	. . . . . . . . . 10 |- ( { <. B , y >. } ` B ) = y
23	20, 22	eqtrdi 2226	. . . . . . . . 9 |- ( z = { <. B , y >. } -> ( z ` B ) = y )
24	23	eqeq2d 2189	. . . . . . . 8 |- ( z = { <. B , y >. } -> ( w = ( z ` B ) <-> w = y ) )
25		equcom 1706	. . . . . . . 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 1605	. . 3 |- ( E. y ( y e. A /\ ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) <-> E. y ( y = w /\ ( y e. A /\ z = { <. B , y >. } ) ) )
33		eleq1w 2238	. . . . 5 |- ( y = w -> ( y e. A <-> w e. A ) )
34		opeq2 3777	. . . . . . 7 |- ( y = w -> <. B , y >. = <. B , w >. )
35	34	sneqd 3604	. . . . . 6 |- ( y = w -> { <. B , y >. } = { <. B , w >. } )
36	35	eqeq2d 2189	. . . . 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 2776	. . 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 6764	1 |- ( A ^m { B } ) ~~ A

Syntax hints:   /\ wa 104   = wceq 1353   E.wex 1492   e. wcel 2148   E.wrex 2456   _Vcvv 2737   {csn 3591   <.cop 3594   class class class wbr 4000   X. cxp 4621   Fn wfn 5207   ` cfv 5212  (class class class)co 5869   ^m cmap 6642   ~~ cen 6732

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-map 6644  df-en 6735

This theorem is referenced by: (None)