Theorem map1 4417

Description: Set exponentiation: ordinal 1 to any set is equinumerous to ordinal 1. Exercise 4.42(b) of [Mendelson] p. 255.

Hypothesis

Ref	Expression
map1.1	|- A e. V

Assertion

Ref	Expression
map1	|- (1o ^m A) ~~ 1o

Proof of Theorem map1

Step	Hyp	Ref	Expression
1		oprex 3974	. 2 |- (1o ^m A) e. V
2		0ex 2706	. . 3 |- (/) e. V
3	2	a1i 8	. 2 |- (x e. (1o ^m A) -> (/) e. V)
4		map1.1	. . . 4 |- A e. V
5		p0ex 2765	. . . 4 |- {(/)} e. V
6	4, 5	xpex 3255	. . 3 |- (A X. {(/)}) e. V
7	6	a1i 8	. 2 |- (y e. 1o -> (A X. {(/)}) e. V)
8		ancom 435	. . 3 |- ((y e. 1o /\ x = (A X. {(/)})) <-> (x = (A X. {(/)}) /\ y e. 1o))
9		df1o2 4130	. . . . . . 7 |- 1o = {(/)}
10	9	opreq1i 3962	. . . . . 6 |- (1o ^m A) = ({(/)} ^m A)
11	10	eleq2i 1535	. . . . 5 |- (x e. (1o ^m A) <-> x e. ({(/)} ^m A))
12	5, 4	elmap 4324	. . . . 5 |- (x e. ({(/)} ^m A) <-> x:A-->{(/)})
13	2	fconst2 3838	. . . . 5 |- (x:A-->{(/)} <-> x = (A X. {(/)}))
14	11, 12, 13	3bitrr 178	. . . 4 |- (x = (A X. {(/)}) <-> x e. (1o ^m A))
15		el1o 4136	. . . 4 |- (y e. 1o <-> y = (/))
16	14, 15	anbi12i 482	. . 3 |- ((x = (A X. {(/)}) /\ y e. 1o) <-> (x e. (1o ^m A) /\ y = (/)))
17	8, 16	bitr2 174	. 2 |- ((x e. (1o ^m A) /\ y = (/)) <-> (y e. 1o /\ x = (A X. {(/)})))
18	1, 3, 7, 17	en2 4389	1 |- (1o ^m A) ~~ 1o

Syntax hints:   /\ wa 223   = wceq 954   e. wcel 956  Vcvv 1807  (/)c0 2276  {csn 2405   class class class wbr 2614   X. cxp 3163  -->wf 3173  (class class class)co 3954  1oc1o 4118   ^m cm 4312   ~~ cen 4354

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-suc 2949  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-opr 3956  df-oprab 3957  df-1o 4123  df-map 4314  df-en 4357

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