Theorem mapsnen 8035

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		ovex 6678	. 2 |- ( A ^m { B } ) e. _V
2		mapsnen.1	. 2 |- A e. _V
3		fvexd 6203	. 2 |- ( z e. ( A ^m { B } ) -> ( z ` B ) e. _V )
4		snex 4908	. . 3 |- { <. B , w >. } e. _V
5	4	a1i 11	. 2 |- ( w e. A -> { <. B , w >. } e. _V )
6		mapsnen.2	. . . . . . 7 |- B e. _V
7	2, 6	mapsn 7899	. . . . . 6 |- ( A ^m { B } ) = { z | E. y e. A z = { <. B , y >. } }
8	7	abeq2i 2735	. . . . 5 |- ( z e. ( A ^m { B } ) <-> E. y e. A z = { <. B , y >. } )
9	8	anbi1i 731	. . . 4 |- ( ( z e. ( A ^m { B } ) /\ w = ( z ` B ) ) <-> ( E. y e. A z = { <. B , y >. } /\ w = ( z ` B ) ) )
10		r19.41v 3089	. . . 4 |- ( E. y e. A ( z = { <. B , y >. } /\ w = ( z ` B ) ) <-> ( E. y e. A z = { <. B , y >. } /\ w = ( z ` B ) ) )
11		df-rex 2918	. . . 4 |- ( E. y e. A ( z = { <. B , y >. } /\ w = ( z ` B ) ) <-> E. y ( y e. A /\ ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) )
12	9, 10, 11	3bitr2i 288	. . 3 |- ( ( z e. ( A ^m { B } ) /\ w = ( z ` B ) ) <-> E. y ( y e. A /\ ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) )
13		fveq1 6190	. . . . . . . . . 10 |- ( z = { <. B , y >. } -> ( z ` B ) = ( { <. B , y >. } ` B ) )
14		vex 3203	. . . . . . . . . . 11 |- y e. _V
15	6, 14	fvsn 6446	. . . . . . . . . 10 |- ( { <. B , y >. } ` B ) = y
16	13, 15	syl6eq 2672	. . . . . . . . 9 |- ( z = { <. B , y >. } -> ( z ` B ) = y )
17	16	eqeq2d 2632	. . . . . . . 8 |- ( z = { <. B , y >. } -> ( w = ( z ` B ) <-> w = y ) )
18		equcom 1945	. . . . . . . 8 |- ( w = y <-> y = w )
19	17, 18	syl6bb 276	. . . . . . 7 |- ( z = { <. B , y >. } -> ( w = ( z ` B ) <-> y = w ) )
20	19	pm5.32i 669	. . . . . 6 |- ( ( z = { <. B , y >. } /\ w = ( z ` B ) ) <-> ( z = { <. B , y >. } /\ y = w ) )
21	20	anbi2i 730	. . . . 5 |- ( ( y e. A /\ ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) <-> ( y e. A /\ ( z = { <. B , y >. } /\ y = w ) ) )
22		anass 681	. . . . 5 |- ( ( ( y e. A /\ z = { <. B , y >. } ) /\ y = w ) <-> ( y e. A /\ ( z = { <. B , y >. } /\ y = w ) ) )
23		ancom 466	. . . . 5 |- ( ( ( y e. A /\ z = { <. B , y >. } ) /\ y = w ) <-> ( y = w /\ ( y e. A /\ z = { <. B , y >. } ) ) )
24	21, 22, 23	3bitr2i 288	. . . 4 |- ( ( y e. A /\ ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) <-> ( y = w /\ ( y e. A /\ z = { <. B , y >. } ) ) )
25	24	exbii 1774	. . 3 |- ( E. y ( y e. A /\ ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) <-> E. y ( y = w /\ ( y e. A /\ z = { <. B , y >. } ) ) )
26		vex 3203	. . . 4 |- w e. _V
27		eleq1 2689	. . . . 5 |- ( y = w -> ( y e. A <-> w e. A ) )
28		opeq2 4403	. . . . . . 7 |- ( y = w -> <. B , y >. = <. B , w >. )
29	28	sneqd 4189	. . . . . 6 |- ( y = w -> { <. B , y >. } = { <. B , w >. } )
30	29	eqeq2d 2632	. . . . 5 |- ( y = w -> ( z = { <. B , y >. } <-> z = { <. B , w >. } ) )
31	27, 30	anbi12d 747	. . . 4 |- ( y = w -> ( ( y e. A /\ z = { <. B , y >. } ) <-> ( w e. A /\ z = { <. B , w >. } ) ) )
32	26, 31	ceqsexv 3242	. . 3 |- ( E. y ( y = w /\ ( y e. A /\ z = { <. B , y >. } ) ) <-> ( w e. A /\ z = { <. B , w >. } ) )
33	12, 25, 32	3bitri 286	. 2 |- ( ( z e. ( A ^m { B } ) /\ w = ( z ` B ) ) <-> ( w e. A /\ z = { <. B , w >. } ) )
34	1, 2, 3, 5, 33	en2i 7993	1 |- ( A ^m { B } ) ~~ A

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   {csn 4177   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   ~~ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-en 7956

This theorem is referenced by:  map2xp  8130  mapdom3  8132  ackbij1lem5  9046  pwxpndom2  9487  hashmap  13222