Theorem map1 7091

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 6919	. . 3 |- ^m Fn ( _V X. _V )
2		1oex 6685	. . 3 |- 1o e. _V
3		elex 2833	. . 3 |- ( A e. V -> A e. _V )
4		fnovex 6108	. . 3 |- ( ( ^m Fn ( _V X. _V ) /\ 1o e. _V /\ A e. _V ) -> ( 1o ^m A ) e. _V )
5	1, 2, 3, 4	mp3an12i 1382	. 2 |- ( A e. V -> ( 1o ^m A ) e. _V )
6	2	a1i 9	. 2 |- ( A e. V -> 1o e. _V )
7		0ex 4255	. . 3 |- (/) e. _V
8	7	2a1i 27	. 2 |- ( A e. V -> ( x e. ( 1o ^m A ) -> (/) e. _V ) )
9		p0ex 4320	. . . 4 |- { (/) } e. _V
10		xpexg 4884	. . . 4 |- ( ( A e. V /\ { (/) } e. _V ) -> ( A X. { (/) } ) e. _V )
11	9, 10	mpan2 429	. . 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 6700	. . . . 5 |- ( y e. 1o <-> y = (/) )
14	13	a1i 9	. . . 4 |- ( A e. V -> ( y e. 1o <-> y = (/) ) )
15		df1o2 6691	. . . . . . . 8 |- 1o = { (/) }
16	15	oveq1i 6085	. . . . . . 7 |- ( 1o ^m A ) = ( { (/) } ^m A )
17	16	eleq2i 2305	. . . . . 6 |- ( x e. ( 1o ^m A ) <-> x e. ( { (/) } ^m A ) )
18		elmapg 6925	. . . . . . 7 |- ( ( { (/) } e. _V /\ A e. V ) -> ( x e. ( { (/) } ^m A ) <-> x : A --> { (/) } ) )
19	9, 18	mpan 428	. . . . . 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 5923	. . . . 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 477	. . 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 7044	1 |- ( A e. V -> ( 1o ^m A ) ~~ 1o )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   (/)c0 3520   {csn 3705   class class class wbr 4125   X. cxp 4767   Fn wfn 5367   -->wf 5368  (class class class)co 6075   1oc1o 6670   ^m cmap 6912   ~~ cen 7010

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 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

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-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-1o 6677  df-map 6914  df-en 7013

This theorem is referenced by: (None)