Theorem map0e 3266

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 ∈ V

Assertion

Ref	Expression
map0e	⊢ (A ↑m ∅) = 1o

Proof of Theorem map0e

Step	Hyp	Ref	Expression
1		fn0 2739	. . . . . 6 ⊢ (f Fn ∅ ↔ f = ∅)
2	1	anbi1i 368	. . . . 5 ⊢ ((f Fn ∅ ∧ ran f ⊆ A) ↔ (f = ∅ ∧ ran f ⊆ A))
3		df-f 2434	. . . . 5 ⊢ (f:∅–→A ↔ (f Fn ∅ ∧ ran f ⊆ A))
4		0ss 1725	. . . . . . 7 ⊢ ∅ ⊆ A
5		rneq 2555	. . . . . . . . 9 ⊢ (f = ∅ → ran f = ran ∅)
6		rn0 2567	. . . . . . . . 9 ⊢ ran ∅ = ∅
7	5, 6	syl6eq 1140	. . . . . . . 8 ⊢ (f = ∅ → ran f = ∅)
8	7	sseq1d 1527	. . . . . . 7 ⊢ (f = ∅ → (ran f ⊆ A ↔ ∅ ⊆ A))
9	4, 8	mpbiri 169	. . . . . 6 ⊢ (f = ∅ → ran f ⊆ A)
10	9	pm4.71i 483	. . . . 5 ⊢ (f = ∅ ↔ (f = ∅ ∧ ran f ⊆ A))
11	2, 3, 10	3bitr4 158	. . . 4 ⊢ (f:∅–→A ↔ f = ∅)
12	11	biabi 1181	. . 3 ⊢ {f∣f:∅–→A} = {f∣f = ∅}
13		map0e.1	. . . 4 ⊢ A ∈ V
14		0ex 1745	. . . 4 ⊢ ∅ ∈ V
15	13, 14	mapval 3264	. . 3 ⊢ (A ↑m ∅) = {f∣f:∅–→A}
16		df-sn 1811	. . 3 ⊢ {∅} = {f∣f = ∅}
17	12, 15, 16	3eqtr4 1126	. 2 ⊢ (A ↑m ∅) = {∅}
18		df1o2 3111	. 2 ⊢ 1o = {∅}
19	17, 18	eqtr4 1122	1 ⊢ (A ↑m ∅) = 1o

Syntax hints:   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ∅c0 1707  {csn 1808  ran crn 2411   Fn wfn 2417  –→wf 2418  (class class class)co 3001  1_oc1o 3099   ↑_m cm 3258

This theorem is referenced by:  map0 3268  infmap2 4953

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-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-fv 2438  df-opr 3003  df-oprab 3004  df-1o 3104  df-map 3259

metamath.org