Theorem map1 6982

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 6819	. . 3 |- ^m Fn ( _V X. _V )
2		1oex 6585	. . 3 |- 1o e. _V
3		elex 2812	. . 3 |- ( A e. V -> A e. _V )
4		fnovex 6046	. . 3 |- ( ( ^m Fn ( _V X. _V ) /\ 1o e. _V /\ A e. _V ) -> ( 1o ^m A ) e. _V )
5	1, 2, 3, 4	mp3an12i 1375	. 2 |- ( A e. V -> ( 1o ^m A ) e. _V )
6	2	a1i 9	. 2 |- ( A e. V -> 1o e. _V )
7		0ex 4214	. . 3 |- (/) e. _V
8	7	2a1i 27	. 2 |- ( A e. V -> ( x e. ( 1o ^m A ) -> (/) e. _V ) )
9		p0ex 4276	. . . 4 |- { (/) } e. _V
10		xpexg 4838	. . . 4 |- ( ( A e. V /\ { (/) } e. _V ) -> ( A X. { (/) } ) e. _V )
11	9, 10	mpan2 425	. . 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 6600	. . . . 5 |- ( y e. 1o <-> y = (/) )
14	13	a1i 9	. . . 4 |- ( A e. V -> ( y e. 1o <-> y = (/) ) )
15		df1o2 6591	. . . . . . . 8 |- 1o = { (/) }
16	15	oveq1i 6023	. . . . . . 7 |- ( 1o ^m A ) = ( { (/) } ^m A )
17	16	eleq2i 2296	. . . . . 6 |- ( x e. ( 1o ^m A ) <-> x e. ( { (/) } ^m A ) )
18		elmapg 6825	. . . . . . 7 |- ( ( { (/) } e. _V /\ A e. V ) -> ( x e. ( { (/) } ^m A ) <-> x : A --> { (/) } ) )
19	9, 18	mpan 424	. . . . . 6 |- ( A e. V -> ( x e. ( { (/) } ^m A ) <-> x : A --> { (/) } ) )
20	17, 19	bitrid 192	. . . . 5 |- ( A e. V -> ( x e. ( 1o ^m A ) <-> x : A --> { (/) } ) )
21	7	fconst2 5866	. . . . 5 |- ( x : A --> { (/) } <-> x = ( A X. { (/) } ) )
22	20, 21	bitr2di 197	. . . 4 |- ( A e. V -> ( x = ( A X. { (/) } ) <-> x e. ( 1o ^m A ) ) )
23	14, 22	anbi12d 473	. . 3 |- ( A e. V -> ( ( y e. 1o /\ x = ( A X. { (/) } ) ) <-> ( y = (/) /\ x e. ( 1o ^m A ) ) ) )
24		ancom 266	. . 3 |- ( ( y = (/) /\ x e. ( 1o ^m A ) ) <-> ( x e. ( 1o ^m A ) /\ y = (/) ) )
25	23, 24	bitr2di 197	. 2 |- ( A e. V -> ( ( x e. ( 1o ^m A ) /\ y = (/) ) <-> ( y e. 1o /\ x = ( A X. { (/) } ) ) ) )
26	5, 6, 8, 12, 25	en2d 6936	1 |- ( A e. V -> ( 1o ^m A ) ~~ 1o )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2800   (/)c0 3492   {csn 3667   class class class wbr 4086   X. cxp 4721   Fn wfn 5319   -->wf 5320  (class class class)co 6013   1oc1o 6570   ^m cmap 6812   ~~ cen 6902

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-1o 6577  df-map 6814  df-en 6905

This theorem is referenced by: (None)