Theorem map1 3335

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 3018	. 2 |- (1o ^m A) e. V
2		0ex 1745	. . 3 |- (/) e. V
3	2	a1i 7	. 2 |- (x e. (1o ^m A) -> (/) e. V)
4		map1.1	. . . 4 |- A e. V
5		p0ex 1885	. . . 4 |- {(/)} e. V
6	4, 5	xpex 2488	. . 3 |- (A X. {(/)}) e. V
7	6	a1i 7	. 2 |- (y e. 1o -> (A X. {(/)}) e. V)
8		ancom 333	. . 3 |- ((y e. 1o /\ x = (A X. {(/)})) <-> (x = (A X. {(/)}) /\ y e. 1o))
9		df1o2 3111	. . . . . . . 8 |- 1o = {(/)}
10	9	opreq1i 3009	. . . . . . 7 |- (1o ^m A) = ({(/)} ^m A)
11	10	eleq2i 1153	. . . . . 6 |- (x e. (1o ^m A) <-> x e. ({(/)} ^m A))
12	5, 4	elmap 3265	. . . . . 6 |- (x e. ({(/)} ^m A) <-> x:A-->{(/)})
13	11, 12	bitr 151	. . . . 5 |- (x e. (1o ^m A) <-> x:A-->{(/)})
14	2	fconst2 2902	. . . . 5 |- (x:A-->{(/)} <-> x = (A X. {(/)}))
15	13, 14	bitr2 152	. . . 4 |- (x = (A X. {(/)}) <-> x e. (1o ^m A))
16	9	eleq2i 1153	. . . . 5 |- (y e. 1o <-> y e. {(/)})
17		elsn 1820	. . . . 5 |- (y e. {(/)} <-> y = (/))
18	16, 17	bitr 151	. . . 4 |- (y e. 1o <-> y = (/))
19	15, 18	anbi12i 369	. . 3 |- ((x = (A X. {(/)}) /\ y e. 1o) <-> (x e. (1o ^m A) /\ y = (/)))
20	8, 19	bitr2 152	. 2 |- ((x e. (1o ^m A) /\ y = (/)) <-> (y e. 1o /\ x = (A X. {(/)})))
21	1, 3, 7, 20	en2 3305	1 |- (1o ^m A) ~~ 1o

Syntax hints:   /\ wa 196   = wceq 1091   e. wcel 1092  Vcvv 1348  (/)c0 1707  {csn 1808   class class class wbr 2054   X. cxp 2408  -->wf 2418  (class class class)co 3001  1oc1o 3099   ^m cm 3258   ~~ cen 3271

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-opr 3003  df-oprab 3004  df-1o 3104  df-map 3259  df-en 3274

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