Theorem map0e 4332

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 3597	. . . . . 6 |- (f Fn (/) <-> f = (/))
2	1	anbi1i 481	. . . . 5 |- ((f Fn (/) /\ ran f (_ A) <-> (f = (/) /\ ran f (_ A))
3		df-f 3189	. . . . 5 |- (f:(/)-->A <-> (f Fn (/) /\ ran f (_ A))
4		0ss 2297	. . . . . . 7 |- (/) (_ A
5		rneq 3334	. . . . . . . . 9 |- (f = (/) -> ran f = ran (/))
6		rn0 3349	. . . . . . . . 9 |- ran (/) = (/)
7	5, 6	syl6eq 1520	. . . . . . . 8 |- (f = (/) -> ran f = (/))
8	7	sseq1d 2084	. . . . . . 7 |- (f = (/) -> (ran f (_ A <-> (/) (_ A))
9	4, 8	mpbiri 194	. . . . . 6 |- (f = (/) -> ran f (_ A)
10	9	pm4.71i 636	. . . . 5 |- (f = (/) <-> (f = (/) /\ ran f (_ A))
11	2, 3, 10	3bitr4 183	. . . 4 |- (f:(/)-->A <-> f = (/))
12	11	abbii 1572	. . 3 |- {f | f:(/)-->A} = {f | f = (/)}
13		map0e.1	. . . 4 |- A e. V
14		0ex 2706	. . . 4 |- (/) e. V
15	13, 14	mapval 4322	. . 3 |- (A ^m (/)) = {f | f:(/)-->A}
16		df-sn 2408	. . 3 |- {(/)} = {f | f = (/)}
17	12, 15, 16	3eqtr4 1502	. 2 |- (A ^m (/)) = {(/)}
18		df1o2 4130	. 2 |- 1o = {(/)}
19	17, 18	eqtr4 1495	1 |- (A ^m (/)) = 1o

Syntax hints:   /\ wa 223   = wceq 954   e. wcel 956  {cab 1461  Vcvv 1807   (_ wss 2043  (/)c0 2276  {csn 2405  ran crn 3166   Fn wfn 3172  -->wf 3173  (class class class)co 3954  1oc1o 4118   ^m cm 4312

This theorem is referenced by:  map0 4334  infmap2 7531

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-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-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-fv 3193  df-opr 3956  df-oprab 3957  df-1o 4123  df-map 4314

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