Theorem mapsnend 7054

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 6891	. . 3 |- ^m Fn ( _V X. _V )
2		mapsnend.a	. . . 4 |- ( ph -> A e. V )
3	2	elexd 2829	. . 3 |- ( ph -> A e. _V )
4		mapsnend.b	. . . 4 |- ( ph -> B e. W )
5		snexg 4299	. . . 4 |- ( B e. W -> { B } e. _V )
6	4, 5	syl 14	. . 3 |- ( ph -> { B } e. _V )
7		fnovex 6085	. . 3 |- ( ( ^m Fn ( _V X. _V ) /\ A e. _V /\ { B } e. _V ) -> ( A ^m { B } ) e. _V )
8	1, 3, 6, 7	mp3an2i 1379	. 2 |- ( ph -> ( A ^m { B } ) e. _V )
9		vex 2818	. . . 4 |- z e. _V
10		fvexg 5691	. . . 4 |- ( ( z e. _V /\ B e. W ) -> ( z ` B ) e. _V )
11	9, 4, 10	sylancr 414	. . 3 |- ( ph -> ( z ` B ) e. _V )
12	11	a1d 22	. 2 |- ( ph -> ( z e. ( A ^m { B } ) -> ( z ` B ) e. _V ) )
13		vex 2818	. . . . 5 |- w e. _V
14		opexg 4346	. . . . 5 |- ( ( B e. W /\ w e. _V ) -> <. B , w >. e. _V )
15	4, 13, 14	sylancl 413	. . . 4 |- ( ph -> <. B , w >. e. _V )
16		snexg 4299	. . . 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 6925	. . . . . 6 |- ( ph -> ( A ^m { B } ) = { z | E. y e. A z = { <. B , y >. } } )
20	19	eqabrd 2372	. . . . 5 |- ( ph -> ( z e. ( A ^m { B } ) <-> E. y e. A z = { <. B , y >. } ) )
21	20	anbi1d 465	. . . 4 |- ( ph -> ( ( z e. ( A ^m { B } ) /\ w = ( z ` B ) ) <-> ( E. y e. A z = { <. B , y >. } /\ w = ( z ` B ) ) ) )
22		r19.41v 2701	. . . . . 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 2528	. . . . 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 5671	. . . . . . . . . 10 |- ( z = { <. B , y >. } -> ( z ` B ) = ( { <. B , y >. } ` B ) )
29		vex 2818	. . . . . . . . . . 11 |- y e. _V
30		fvsng 5882	. . . . . . . . . . 11 |- ( ( B e. W /\ y e. _V ) -> ( { <. B , y >. } ` B ) = y )
31	4, 29, 30	sylancl 413	. . . . . . . . . 10 |- ( ph -> ( { <. B , y >. } ` B ) = y )
32	28, 31	sylan9eqr 2289	. . . . . . . . 9 |- ( ( ph /\ z = { <. B , y >. } ) -> ( z ` B ) = y )
33	32	eqeq2d 2246	. . . . . . . 8 |- ( ( ph /\ z = { <. B , y >. } ) -> ( w = ( z ` B ) <-> w = y ) )
34		equcom 1754	. . . . . . . 8 |- ( w = y <-> y = w )
35	33, 34	bitrdi 196	. . . . . . 7 |- ( ( ph /\ z = { <. B , y >. } ) -> ( w = ( z ` B ) <-> y = w ) )
36	35	pm5.32da 452	. . . . . 6 |- ( ph -> ( ( z = { <. B , y >. } /\ w = ( z ` B ) ) <-> ( z = { <. B , y >. } /\ y = w ) ) )
37	36	anbi2d 464	. . . . 5 |- ( ph -> ( ( y e. A /\ ( z = { <. B , y >. } /\ w = ( z ` B ) ) ) <-> ( y e. A /\ ( z = { <. B , y >. } /\ y = w ) ) ) )
38		anass 401	. . . . . 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 1874	. . 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 2295	. . . . . 6 |- ( y = w -> ( y e. A <-> w e. A ) )
45		opeq2 3886	. . . . . . . 8 |- ( y = w -> <. B , y >. = <. B , w >. )
46	45	sneqd 3704	. . . . . . 7 |- ( y = w -> { <. B , y >. } = { <. B , w >. } )
47	46	eqeq2d 2246	. . . . . 6 |- ( y = w -> ( z = { <. B , y >. } <-> z = { <. B , w >. } ) )
48	44, 47	anbi12d 473	. . . . 5 |- ( y = w -> ( ( y e. A /\ z = { <. B , y >. } ) <-> ( w e. A /\ z = { <. B , w >. } ) ) )
49	48	equsexvw 1777	. . . 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 7009	1 |- ( ph -> ( A ^m { B } ) ~~ A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2205   E.wrex 2523   _Vcvv 2815   {csn 3691   <.cop 3694   class class class wbr 4111   X. cxp 4749   Fn wfn 5349   ` cfv 5354  (class class class)co 6052   ^m cmap 6884   ~~ cen 6975

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-map 6886  df-en 6978

This theorem is referenced by:  mapfi  7216  hashmap  11196