Theorem map0e 5401

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 4532	. . . . . 6 |- (f Fn (/) <-> f = (/))
2	1	anbi1i 539	. . . . 5 |- ((f Fn (/) /\ ran f C_ A) <-> (f = (/) /\ ran f C_ A))
3		df-f 4010	. . . . 5 |- (f:(/)-->A <-> (f Fn (/) /\ ran f C_ A))
4		0ss 2900	. . . . . . 7 |- (/) C_ A
5		rneq 4186	. . . . . . . . 9 |- (f = (/) -> ran f = ran (/))
6		rn0 4203	. . . . . . . . 9 |- ran (/) = (/)
7	5, 6	syl6eq 1944	. . . . . . . 8 |- (f = (/) -> ran f = (/))
8	7	sseq1d 2644	. . . . . . 7 |- (f = (/) -> (ran f C_ A <-> (/) C_ A))
9	4, 8	mpbiri 211	. . . . . 6 |- (f = (/) -> ran f C_ A)
10	9	pm4.71i 699	. . . . 5 |- (f = (/) <-> (f = (/) /\ ran f C_ A))
11	2, 3, 10	3bitr4i 200	. . . 4 |- (f:(/)-->A <-> f = (/))
12	11	abbii 2006	. . 3 |- {f | f:(/)-->A} = {f | f = (/)}
13		map0e.1	. . . 4 |- A e. _V
14		0ex 3446	. . . 4 |- (/) e. _V
15	13, 14	mapval 5391	. . 3 |- (A ^m (/)) = {f | f:(/)-->A}
16		df-sn 3049	. . 3 |- {(/)} = {f | f = (/)}
17	12, 15, 16	3eqtr4i 1921	. 2 |- (A ^m (/)) = {(/)}
18		df1o2 5185	. 2 |- 1o = {(/)}
19	17, 18	eqtr4i 1911	1 |- (A ^m (/)) = 1o

Syntax hints:   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  _Vcvv 2292   C_ wss 2593  (/)c0 2875  {csn 3044  ran crn 3987   Fn wfn 3993  -->wf 3994  (class class class)co 4884  1oc1o 5172   ^m cmap 5381

This theorem is referenced by:  map0 5403  infmap2 8850

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-1o 5177  df-map 5383

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