Theorem map1 6788

Description: Set exponentiation: ordinal 1 to any set is equinumerous to ordinal 1. Exercise 4.42(b) of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.)

Assertion

Ref	Expression
map1	|- ( A e. V -> ( 1o ^m A ) ~~ 1o )

Proof of Theorem map1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnmap 6631	. . 3 |- ^m Fn ( _V X. _V )
2		1oex 6401	. . 3 |- 1o e. _V
3		elex 2741	. . 3 |- ( A e. V -> A e. _V )
4		fnovex 5884	. . 3 |- ( ( ^m Fn ( _V X. _V ) /\ 1o e. _V /\ A e. _V ) -> ( 1o ^m A ) e. _V )
5	1, 2, 3, 4	mp3an12i 1336	. 2 |- ( A e. V -> ( 1o ^m A ) e. _V )
6	2	a1i 9	. 2 |- ( A e. V -> 1o e. _V )
7		0ex 4114	. . 3 |- (/) e. _V
8	7	2a1i 27	. 2 |- ( A e. V -> ( x e. ( 1o ^m A ) -> (/) e. _V ) )
9		p0ex 4172	. . . 4 |- { (/) } e. _V
10		xpexg 4723	. . . 4 |- ( ( A e. V /\ { (/) } e. _V ) -> ( A X. { (/) } ) e. _V )
11	9, 10	mpan2 423	. . 3 |- ( A e. V -> ( A X. { (/) } ) e. _V )
12	11	a1d 22	. 2 |- ( A e. V -> ( y e. 1o -> ( A X. { (/) } ) e. _V ) )
13		el1o 6414	. . . . 5 |- ( y e. 1o <-> y = (/) )
14	13	a1i 9	. . . 4 |- ( A e. V -> ( y e. 1o <-> y = (/) ) )
15		df1o2 6406	. . . . . . . 8 |- 1o = { (/) }
16	15	oveq1i 5861	. . . . . . 7 |- ( 1o ^m A ) = ( { (/) } ^m A )
17	16	eleq2i 2237	. . . . . 6 |- ( x e. ( 1o ^m A ) <-> x e. ( { (/) } ^m A ) )
18		elmapg 6637	. . . . . . 7 |- ( ( { (/) } e. _V /\ A e. V ) -> ( x e. ( { (/) } ^m A ) <-> x : A --> { (/) } ) )
19	9, 18	mpan 422	. . . . . 6 |- ( A e. V -> ( x e. ( { (/) } ^m A ) <-> x : A --> { (/) } ) )
20	17, 19	syl5bb 191	. . . . 5 |- ( A e. V -> ( x e. ( 1o ^m A ) <-> x : A --> { (/) } ) )
21	7	fconst2 5711	. . . . 5 |- ( x : A --> { (/) } <-> x = ( A X. { (/) } ) )
22	20, 21	bitr2di 196	. . . 4 |- ( A e. V -> ( x = ( A X. { (/) } ) <-> x e. ( 1o ^m A ) ) )
23	14, 22	anbi12d 470	. . 3 |- ( A e. V -> ( ( y e. 1o /\ x = ( A X. { (/) } ) ) <-> ( y = (/) /\ x e. ( 1o ^m A ) ) ) )
24		ancom 264	. . 3 |- ( ( y = (/) /\ x e. ( 1o ^m A ) ) <-> ( x e. ( 1o ^m A ) /\ y = (/) ) )
25	23, 24	bitr2di 196	. 2 |- ( A e. V -> ( ( x e. ( 1o ^m A ) /\ y = (/) ) <-> ( y e. 1o /\ x = ( A X. { (/) } ) ) ) )
26	5, 6, 8, 12, 25	en2d 6744	1 |- ( A e. V -> ( 1o ^m A ) ~~ 1o )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   _Vcvv 2730   (/)c0 3414   {csn 3581   class class class wbr 3987   X. cxp 4607   Fn wfn 5191   -->wf 5192  (class class class)co 5851   1oc1o 6386   ^m cmap 6624   ~~ cen 6714

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-suc 4354  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-1o 6393  df-map 6626  df-en 6717

This theorem is referenced by: (None)