Theorem map0b 3267

Description: Set exponentiation with an empty base is the empty set, provided the exponent is non-empty. Theorem 96 of [Suppes] p. 89.

Hypothesis

Ref	Expression
map0e.1	⊢ A ∈ V

Assertion

Ref	Expression
map0b	⊢ (¬ A = ∅ → (∅ ↑m A) = ∅)

Proof of Theorem map0b

Step	Hyp	Ref	Expression
1		abn0 1715	. . . 4 ⊢ (¬ {f∣f:A–→∅} = ∅ ↔ ∃f f:A–→∅)
2		fdm 2756	. . . . . 6 ⊢ (f:A–→∅ → dom f = A)
3		frn 2757	. . . . . . . 8 ⊢ (f:A–→∅ → ran f ⊆ ∅)
4		ss0 1727	. . . . . . . 8 ⊢ (ran f ⊆ ∅ → ran f = ∅)
5	3, 4	syl 12	. . . . . . 7 ⊢ (f:A–→∅ → ran f = ∅)
6		dm0rn0 2549	. . . . . . 7 ⊢ (dom f = ∅ ↔ ran f = ∅)
7	5, 6	sylibr 175	. . . . . 6 ⊢ (f:A–→∅ → dom f = ∅)
8	2, 7	eqtr3d 1130	. . . . 5 ⊢ (f:A–→∅ → A = ∅)
9	8	19.23aiv 952	. . . 4 ⊢ (∃f f:A–→∅ → A = ∅)
10	1, 9	sylbi 174	. . 3 ⊢ (¬ {f∣f:A–→∅} = ∅ → A = ∅)
11	10	con1i 88	. 2 ⊢ (¬ A = ∅ → {f∣f:A–→∅} = ∅)
12		0ex 1745	. . 3 ⊢ ∅ ∈ V
13		map0e.1	. . 3 ⊢ A ∈ V
14	12, 13	mapval 3264	. 2 ⊢ (∅ ↑m A) = {f∣f:A–→∅}
15	11, 14	syl5eq 1136	1 ⊢ (¬ A = ∅ → (∅ ↑m A) = ∅)

Syntax hints:  ¬ wn 1   → wi 2  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ∅c0 1707  dom cdm 2410  ran crn 2411  –→wf 2418  (class class class)co 3001   ↑_m cm 3258

This theorem is referenced by:  map0 3268  mapdom2 3389

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-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-map 3259

metamath.org