Theorem map1 6928

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 6765	. . 3 |- ^m Fn ( _V X. _V )
2		1oex 6533	. . 3 |- 1o e. _V
3		elex 2788	. . 3 |- ( A e. V -> A e. _V )
4		fnovex 6000	. . 3 |- ( ( ^m Fn ( _V X. _V ) /\ 1o e. _V /\ A e. _V ) -> ( 1o ^m A ) e. _V )
5	1, 2, 3, 4	mp3an12i 1354	. 2 |- ( A e. V -> ( 1o ^m A ) e. _V )
6	2	a1i 9	. 2 |- ( A e. V -> 1o e. _V )
7		0ex 4187	. . 3 |- (/) e. _V
8	7	2a1i 27	. 2 |- ( A e. V -> ( x e. ( 1o ^m A ) -> (/) e. _V ) )
9		p0ex 4248	. . . 4 |- { (/) } e. _V
10		xpexg 4807	. . . 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 6546	. . . . 5 |- ( y e. 1o <-> y = (/) )
14	13	a1i 9	. . . 4 |- ( A e. V -> ( y e. 1o <-> y = (/) ) )
15		df1o2 6538	. . . . . . . 8 |- 1o = { (/) }
16	15	oveq1i 5977	. . . . . . 7 |- ( 1o ^m A ) = ( { (/) } ^m A )
17	16	eleq2i 2274	. . . . . 6 |- ( x e. ( 1o ^m A ) <-> x e. ( { (/) } ^m A ) )
18		elmapg 6771	. . . . . . 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 5824	. . . . 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 6882	1 |- ( A e. V -> ( 1o ^m A ) ~~ 1o )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   _Vcvv 2776   (/)c0 3468   {csn 3643   class class class wbr 4059   X. cxp 4691   Fn wfn 5285   -->wf 5286  (class class class)co 5967   1oc1o 6518   ^m cmap 6758   ~~ cen 6848

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-1o 6525  df-map 6760  df-en 6851

This theorem is referenced by: (None)