Theorem map0e 4342

Description: Set exponentiation with an empty exponent (ordinal number 0) is ordinal number 1. Exercise 4.42(a) of [Mendelson] p. 255.

Hypothesis

Ref	Expression
map0e.1	|- A e. V

Assertion

Ref	Expression
map0e	|- (A ^m (/)) = 1o

Proof of Theorem map0e

Step	Hyp	Ref	Expression
1		fn0 3605	. . . . . 6 |- (f Fn (/) <-> f = (/))
2	1	anbi1i 481	. . . . 5 |- ((f Fn (/) /\ ran f (_ A) <-> (f = (/) /\ ran f (_ A))
3		df-f 3194	. . . . 5 |- (f:(/)-->A <-> (f Fn (/) /\ ran f (_ A))
4		0ss 2301	. . . . . . 7 |- (/) (_ A
5		rneq 3339	. . . . . . . . 9 |- (f = (/) -> ran f = ran (/))
6		rn0 3355	. . . . . . . . 9 |- ran (/) = (/)
7	5, 6	syl6eq 1523	. . . . . . . 8 |- (f = (/) -> ran f = (/))
8	7	sseq1d 2088	. . . . . . 7 |- (f = (/) -> (ran f (_ A <-> (/) (_ A))
9	4, 8	mpbiri 194	. . . . . 6 |- (f = (/) -> ran f (_ A)
10	9	pm4.71i 637	. . . . 5 |- (f = (/) <-> (f = (/) /\ ran f (_ A))
11	2, 3, 10	3bitr4 183	. . . 4 |- (f:(/)-->A <-> f = (/))
12	11	abbii 1575	. . 3 |- {f | f:(/)-->A} = {f | f = (/)}
13		map0e.1	. . . 4 |- A e. V
14		0ex 2711	. . . 4 |- (/) e. V
15	13, 14	mapval 4332	. . 3 |- (A ^m (/)) = {f | f:(/)-->A}
16		df-sn 2412	. . 3 |- {(/)} = {f | f = (/)}
17	12, 15, 16	3eqtr4 1505	. 2 |- (A ^m (/)) = {(/)}
18		df1o2 4140	. 2 |- 1o = {(/)}
19	17, 18	eqtr4 1498	1 |- (A ^m (/)) = 1o

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  Vcvv 1811   (_ wss 2047  (/)c0 2280  {csn 2409  ran crn 3171   Fn wfn 3177  -->wf 3178  (class class class)co 3963  1oc1o 4128   ^m cm 4322

This theorem is referenced by:  map0 4344  infmap2 7581

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-oprab 3966  df-1o 4133  df-map 4324

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