Theorem mapsnend 7092

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.) (Revised by Glauco Siliprandi, 24-Dec-2020.)

Hypotheses

Ref	Expression
mapsnend.a	|- ( ph -> A e. V )
mapsnend.b	|- ( ph -> B e. W )

Assertion

Ref	Expression
mapsnend	|- ( ph -> ( A ^m { B } ) ~~ A )

Proof of Theorem mapsnend

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

Step	Hyp	Ref	Expression
1		fnmap 6922	. . 3 |- ^m Fn ( _V X. _V )
2		mapsnend.a	. . . 4 |- ( ph -> A e. V )
3	2	elexd 2835	. . 3 |- ( ph -> A e. _V )
4		mapsnend.b	. . . 4 |- ( ph -> B e. W )
5		snexg 4319	. . . 4 |- ( B e. W -> { B } e. _V )
6	4, 5	syl 14	. . 3 |- ( ph -> { B } e. _V )
7		fnovex 6111	. . 3 |- ( ( ^m Fn ( _V X. _V ) /\ A e. _V /\ { B } e. _V ) -> ( A ^m { B } ) e. _V )
8	1, 3, 6, 7	mp3an2i 1383	. 2 |- ( ph -> ( A ^m { B } ) e. _V )
9		vex 2824	. . . 4 |- z e. _V
10		fvexg 5712	. . . 4 |- ( ( z e. _V /\ B e. W ) -> ( z ` B ) e. _V )
11	9, 4, 10	sylancr 418	. . 3 |- ( ph -> ( z ` B ) e. _V )
12	11	a1d 22	. 2 |- ( ph -> ( z e. ( A ^m { B } ) -> ( z ` B ) e. _V ) )
13		vex 2824	. . . . 5 |- w e. _V
14		opexg 4366	. . . . 5 |- ( ( B e. W /\ w e. _V ) -> <. B , w >. e. _V )
15	4, 13, 14	sylancl 417	. . . 4 |- ( ph -> <. B , w >. e. _V )
16		snexg 4319	. . . 4 |- ( <. B , w >. e. _V -> { <. B , w >. } e. _V )
17	15, 16	syl 14	. . 3 |- ( ph -> { <. B , w >. } e. _V )
18	17	a1d 22	. 2 |- ( ph -> ( w e. A -> { <. B , w >. } e. _V ) )
19	2, 4	mapsnd 6963	. . . . . 6 |- ( ph -> ( A ^m { B } ) = { z | E. y e. A z = { <. B , y >. } } )
20	19	eqabrd 2378	. . . . 5 |- ( ph -> ( z e. ( A ^m { B } ) <-> E. y e. A z = { <. B , y >. } ) )
21	20	anbi1d 469	. . . 4 |- ( ph -> ( ( z e. ( A ^m { B } ) /\ w = ( z ` B ) ) <-> ( E. y e. A z = { <. B , y >. } /\ w = ( z ` B ) ) ) )
22		r19.41v 2707	. . . . . 6 |- ( E. y e. A ( z = { <. B , y >. } /\ w = ( z ` B ) ) <-> ( E. y e. A z = { <. B , y >. } /\ w = ( z ` B ) ) )
23	22	bicomi 132	. . . . 5 |- ( ( E. y e. A z = { <. B , y >. } /\ w = ( z ` B ) ) <-> E. y e. A ( z = { <. B , y >. } /\ w = ( z ` B ) ) )
24	23	a1i 9	. . . 4 |- ( ph -> ( ( E. y e. A z = { <. B , y >. } /\ w = ( z ` B ) ) <-> E. y e. A ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) )
25		df-rex 2534	. . . . 5 |- ( E. y e. A ( z = { <. B , y >. } /\ w = ( z ` B ) ) <-> E. y ( y e. A /\ ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) )
26	25	a1i 9	. . . 4 |- ( ph -> ( E. y e. A ( z = { <. B , y >. } /\ w = ( z ` B ) ) <-> E. y ( y e. A /\ ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) ) )
27	21, 24, 26	3bitrd 214	. . 3 |- ( ph -> ( ( z e. ( A ^m { B } ) /\ w = ( z ` B ) ) <-> E. y ( y e. A /\ ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) ) )
28		fveq1 5692	. . . . . . . . . 10 |- ( z = { <. B , y >. } -> ( z ` B ) = ( { <. B , y >. } ` B ) )
29		vex 2824	. . . . . . . . . . 11 |- y e. _V
30		fvsng 5905	. . . . . . . . . . 11 |- ( ( B e. W /\ y e. _V ) -> ( { <. B , y >. } ` B ) = y )
31	4, 29, 30	sylancl 417	. . . . . . . . . 10 |- ( ph -> ( { <. B , y >. } ` B ) = y )
32	28, 31	sylan9eqr 2293	. . . . . . . . 9 |- ( ( ph /\ z = { <. B , y >. } ) -> ( z ` B ) = y )
33	32	eqeq2d 2250	. . . . . . . 8 |- ( ( ph /\ z = { <. B , y >. } ) -> ( w = ( z ` B ) <-> w = y ) )
34		equcom 1758	. . . . . . . 8 |- ( w = y <-> y = w )
35	33, 34	bitrdi 196	. . . . . . 7 |- ( ( ph /\ z = { <. B , y >. } ) -> ( w = ( z ` B ) <-> y = w ) )
36	35	pm5.32da 456	. . . . . 6 |- ( ph -> ( ( z = { <. B , y >. } /\ w = ( z ` B ) ) <-> ( z = { <. B , y >. } /\ y = w ) ) )
37	36	anbi2d 468	. . . . 5 |- ( ph -> ( ( y e. A /\ ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) <-> ( y e. A /\ ( z = { <. B , y >. } /\ y = w ) ) ) )
38		anass 405	. . . . . 6 |- ( ( ( y e. A /\ z = { <. B , y >. } ) /\ y = w ) <-> ( y e. A /\ ( z = { <. B , y >. } /\ y = w ) ) )
39	38	a1i 9	. . . . 5 |- ( ph -> ( ( ( y e. A /\ z = { <. B , y >. } ) /\ y = w ) <-> ( y e. A /\ ( z = { <. B , y >. } /\ y = w ) ) ) )
40		ancom 266	. . . . . 6 |- ( ( ( y e. A /\ z = { <. B , y >. } ) /\ y = w ) <-> ( y = w /\ ( y e. A /\ z = { <. B , y >. } ) ) )
41	40	a1i 9	. . . . 5 |- ( ph -> ( ( ( y e. A /\ z = { <. B , y >. } ) /\ y = w ) <-> ( y = w /\ ( y e. A /\ z = { <. B , y >. } ) ) ) )
42	37, 39, 41	3bitr2d 216	. . . 4 |- ( ph -> ( ( y e. A /\ ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) <-> ( y = w /\ ( y e. A /\ z = { <. B , y >. } ) ) ) )
43	42	exbidv 1878	. . 3 |- ( ph -> ( E. y ( y e. A /\ ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) <-> E. y ( y = w /\ ( y e. A /\ z = { <. B , y >. } ) ) ) )
44		eleq1w 2299	. . . . . 6 |- ( y = w -> ( y e. A <-> w e. A ) )
45		opeq2 3903	. . . . . . . 8 |- ( y = w -> <. B , y >. = <. B , w >. )
46	45	sneqd 3721	. . . . . . 7 |- ( y = w -> { <. B , y >. } = { <. B , w >. } )
47	46	eqeq2d 2250	. . . . . 6 |- ( y = w -> ( z = { <. B , y >. } <-> z = { <. B , w >. } ) )
48	44, 47	anbi12d 477	. . . . 5 |- ( y = w -> ( ( y e. A /\ z = { <. B , y >. } ) <-> ( w e. A /\ z = { <. B , w >. } ) ) )
49	48	equsexvw 1781	. . . 4 |- ( E. y ( y = w /\ ( y e. A /\ z = { <. B , y >. } ) ) <-> ( w e. A /\ z = { <. B , w >. } ) )
50	49	a1i 9	. . 3 |- ( ph -> ( E. y ( y = w /\ ( y e. A /\ z = { <. B , y >. } ) ) <-> ( w e. A /\ z = { <. B , w >. } ) ) )
51	27, 43, 50	3bitrd 214	. 2 |- ( ph -> ( ( z e. ( A ^m { B } ) /\ w = ( z ` B ) ) <-> ( w e. A /\ z = { <. B , w >. } ) ) )
52	8, 2, 12, 18, 51	en2d 7047	1 |- ( ph -> ( A ^m { B } ) ~~ A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   E.wrex 2529   _Vcvv 2821   {csn 3708   <.cop 3711   class class class wbr 4128   X. cxp 4770   Fn wfn 5370   ` cfv 5375  (class class class)co 6078   ^m cmap 6915   ~~ cen 7013

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6081  df-oprab 6082  df-mpo 6083  df-1st 6367  df-2nd 6368  df-map 6917  df-en 7016

This theorem is referenced by:  mapfi  7254  hashmap  11249