Theorem map0 3268

Description: Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89.

Hypotheses

Ref	Expression
map0.1	⊢ A ∈ V
map0.2	⊢ B ∈ V

Assertion

Ref	Expression
map0	⊢ ((A ↑m B) = ∅ ↔ (A = ∅ ∧ ¬ B = ∅))

Proof of Theorem map0

Step	Hyp	Ref	Expression
1		map0.1	. . . . . 6 ⊢ A ∈ V
2		map0.2	. . . . . 6 ⊢ B ∈ V
3	1, 2	mapval 3264	. . . . 5 ⊢ (A ↑m B) = {f∣f:B–→A}
4	3	cleq1i 1108	. . . 4 ⊢ ((A ↑m B) = ∅ ↔ {f∣f:B–→A} = ∅)
5		snssi 1851	. . . . . . . 8 ⊢ (x ∈ A → {x} ⊆ A)
6		visset 1350	. . . . . . . . . 10 ⊢ x ∈ V
7	6	fconst 2774	. . . . . . . . 9 ⊢ (B × {x}):B–→{x}
8		fss 2759	. . . . . . . . 9 ⊢ (((B × {x}):B–→{x} ∧ {x} ⊆ A) → (B × {x}):B–→A)
9	7, 8	mpan 518	. . . . . . . 8 ⊢ ({x} ⊆ A → (B × {x}):B–→A)
10		snex 1859	. . . . . . . . . 10 ⊢ {x} ∈ V
11	2, 10	xpex 2488	. . . . . . . . 9 ⊢ (B × {x}) ∈ V
12		feq1 2748	. . . . . . . . 9 ⊢ (f = (B × {x}) → (f:B–→A ↔ (B × {x}):B–→A))
13	11, 12	cla4ev 1401	. . . . . . . 8 ⊢ ((B × {x}):B–→A → ∃f f:B–→A)
14	5, 9, 13	3syl 21	. . . . . . 7 ⊢ (x ∈ A → ∃f f:B–→A)
15	14	19.23aiv 952	. . . . . 6 ⊢ (∃x x ∈ A → ∃f f:B–→A)
16		n0 1714	. . . . . 6 ⊢ (¬ A = ∅ ↔ ∃x x ∈ A)
17		abn0 1715	. . . . . 6 ⊢ (¬ {f∣f:B–→A} = ∅ ↔ ∃f f:B–→A)
18	15, 16, 17	3imtr4 192	. . . . 5 ⊢ (¬ A = ∅ → ¬ {f∣f:B–→A} = ∅)
19	18	a3i 69	. . . 4 ⊢ ({f∣f:B–→A} = ∅ → A = ∅)
20	4, 19	sylbi 174	. . 3 ⊢ ((A ↑m B) = ∅ → A = ∅)
21		0nep0 1887	. . . . . 6 ⊢ ¬ ∅ = {∅}
22	1	map0e 3266	. . . . . . . 8 ⊢ (A ↑m ∅) = 1o
23	22	cleq1i 1108	. . . . . . 7 ⊢ ((A ↑m ∅) = ∅ ↔ 1o = ∅)
24		df1o2 3111	. . . . . . . 8 ⊢ 1o = {∅}
25	24	cleq1i 1108	. . . . . . 7 ⊢ (1o = ∅ ↔ {∅} = ∅)
26		cleqcom 1103	. . . . . . 7 ⊢ ({∅} = ∅ ↔ ∅ = {∅})
27	23, 25, 26	3bitr 155	. . . . . 6 ⊢ ((A ↑m ∅) = ∅ ↔ ∅ = {∅})
28	21, 27	mtbir 167	. . . . 5 ⊢ ¬ (A ↑m ∅) = ∅
29		opreq2 3007	. . . . . 6 ⊢ (B = ∅ → (A ↑m B) = (A ↑m ∅))
30	29	cleq1d 1109	. . . . 5 ⊢ (B = ∅ → ((A ↑m B) = ∅ ↔ (A ↑m ∅) = ∅))
31	28, 30	mtbiri 539	. . . 4 ⊢ (B = ∅ → ¬ (A ↑m B) = ∅)
32	31	con2i 89	. . 3 ⊢ ((A ↑m B) = ∅ → ¬ B = ∅)
33	20, 32	jca 236	. 2 ⊢ ((A ↑m B) = ∅ → (A = ∅ ∧ ¬ B = ∅))
34		opreq1 3006	. . 3 ⊢ (A = ∅ → (A ↑m B) = (∅ ↑m B))
35	2	map0b 3267	. . 3 ⊢ (¬ B = ∅ → (∅ ↑m B) = ∅)
36	34, 35	sylan9eq 1144	. 2 ⊢ ((A = ∅ ∧ ¬ B = ∅) → (A ↑m B) = ∅)
37	33, 36	impbi 139	1 ⊢ ((A ↑m B) = ∅ ↔ (A = ∅ ∧ ¬ B = ∅))

Syntax hints:  ¬ wn 1   ↔ wb 127   ∧ wa 196  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ∅c0 1707  {csn 1808   × cxp 2408  –→wf 2418  (class class class)co 3001  1_oc1o 3099   ↑_m cm 3258

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

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