Theorem mapsnen 6713

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 6557	. . 3 |- ^m Fn ( _V X. _V )
2		mapsnen.1	. . 3 |- A e. _V
3		mapsnen.2	. . . 4 |- B e. _V
4	3	snex 4117	. . 3 |- { B } e. _V
5		fnovex 5812	. . 3 |- ( ( ^m Fn ( _V X. _V ) /\ A e. _V /\ { B } e. _V ) -> ( A ^m { B } ) e. _V )
6	1, 2, 4, 5	mp3an 1316	. 2 |- ( A ^m { B } ) e. _V
7		vex 2692	. . . 4 |- z e. _V
8	7, 3	fvex 5449	. . 3 |- ( z ` B ) e. _V
9	8	a1i 9	. 2 |- ( z e. ( A ^m { B } ) -> ( z ` B ) e. _V )
10		vex 2692	. . . . 5 |- w e. _V
11	3, 10	opex 4159	. . . 4 |- <. B , w >. e. _V
12	11	snex 4117	. . 3 |- { <. B , w >. } e. _V
13	12	a1i 9	. 2 |- ( w e. A -> { <. B , w >. } e. _V )
14	2, 3	mapsn 6592	. . . . . 6 |- ( A ^m { B } ) = { z | E. y e. A z = { <. B , y >. } }
15	14	abeq2i 2251	. . . . 5 |- ( z e. ( A ^m { B } ) <-> E. y e. A z = { <. B , y >. } )
16	15	anbi1i 454	. . . 4 |- ( ( z e. ( A ^m { B } ) /\ w = ( z ` B ) ) <-> ( E. y e. A z = { <. B , y >. } /\ w = ( z ` B ) ) )
17		r19.41v 2590	. . . 4 |- ( E. y e. A ( z = { <. B , y >. } /\ w = ( z ` B ) ) <-> ( E. y e. A z = { <. B , y >. } /\ w = ( z ` B ) ) )
18		df-rex 2423	. . . 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 207	. . 3 |- ( ( z e. ( A ^m { B } ) /\ w = ( z ` B ) ) <-> E. y ( y e. A /\ ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) )
20		fveq1 5428	. . . . . . . . . 10 |- ( z = { <. B , y >. } -> ( z ` B ) = ( { <. B , y >. } ` B ) )
21		vex 2692	. . . . . . . . . . 11 |- y e. _V
22	3, 21	fvsn 5623	. . . . . . . . . 10 |- ( { <. B , y >. } ` B ) = y
23	20, 22	eqtrdi 2189	. . . . . . . . 9 |- ( z = { <. B , y >. } -> ( z ` B ) = y )
24	23	eqeq2d 2152	. . . . . . . 8 |- ( z = { <. B , y >. } -> ( w = ( z ` B ) <-> w = y ) )
25		equcom 1683	. . . . . . . 8 |- ( w = y <-> y = w )
26	24, 25	syl6bb 195	. . . . . . 7 |- ( z = { <. B , y >. } -> ( w = ( z ` B ) <-> y = w ) )
27	26	pm5.32i 450	. . . . . 6 |- ( ( z = { <. B , y >. } /\ w = ( z ` B ) ) <-> ( z = { <. B , y >. } /\ y = w ) )
28	27	anbi2i 453	. . . . 5 |- ( ( y e. A /\ ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) <-> ( y e. A /\ ( z = { <. B , y >. } /\ y = w ) ) )
29		anass 399	. . . . 5 |- ( ( ( y e. A /\ z = { <. B , y >. } ) /\ y = w ) <-> ( y e. A /\ ( z = { <. B , y >. } /\ y = w ) ) )
30		ancom 264	. . . . 5 |- ( ( ( y e. A /\ z = { <. B , y >. } ) /\ y = w ) <-> ( y = w /\ ( y e. A /\ z = { <. B , y >. } ) ) )
31	28, 29, 30	3bitr2i 207	. . . 4 |- ( ( y e. A /\ ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) <-> ( y = w /\ ( y e. A /\ z = { <. B , y >. } ) ) )
32	31	exbii 1585	. . 3 |- ( E. y ( y e. A /\ ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) <-> E. y ( y = w /\ ( y e. A /\ z = { <. B , y >. } ) ) )
33		eleq1w 2201	. . . . 5 |- ( y = w -> ( y e. A <-> w e. A ) )
34		opeq2 3714	. . . . . . 7 |- ( y = w -> <. B , y >. = <. B , w >. )
35	34	sneqd 3545	. . . . . 6 |- ( y = w -> { <. B , y >. } = { <. B , w >. } )
36	35	eqeq2d 2152	. . . . 5 |- ( y = w -> ( z = { <. B , y >. } <-> z = { <. B , w >. } ) )
37	33, 36	anbi12d 465	. . . 4 |- ( y = w -> ( ( y e. A /\ z = { <. B , y >. } ) <-> ( w e. A /\ z = { <. B , w >. } ) ) )
38	10, 37	ceqsexv 2728	. . 3 |- ( E. y ( y = w /\ ( y e. A /\ z = { <. B , y >. } ) ) <-> ( w e. A /\ z = { <. B , w >. } ) )
39	19, 32, 38	3bitri 205	. 2 |- ( ( z e. ( A ^m { B } ) /\ w = ( z ` B ) ) <-> ( w e. A /\ z = { <. B , w >. } ) )
40	6, 2, 9, 13, 39	en2i 6672	1 |- ( A ^m { B } ) ~~ A

Syntax hints:   /\ wa 103   = wceq 1332   E.wex 1469   e. wcel 1481   E.wrex 2418   _Vcvv 2689   {csn 3532   <.cop 3535   class class class wbr 3937   X. cxp 4545   Fn wfn 5126   ` cfv 5131  (class class class)co 5782   ^m cmap 6550   ~~ cen 6640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-map 6552  df-en 6643

This theorem is referenced by: (None)