Theorem map0 6025

Description: Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89. (Contributed by set.mm contributors, 10-Dec-2003.) (Revised by set.mm contributors, 17-May-2007.)

Hypotheses

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

Assertion

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

Proof of Theorem map0

Dummy variables x f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		map0.1	. . . . . 6 ⊢ A ∈ V
2		map0.2	. . . . . 6 ⊢ B ∈ V
3	1, 2	mapval 6011	. . . . 5 ⊢ (A ↑m B) = {f ∣ f:B–→A}
4	3	eqeq1i 2360	. . . 4 ⊢ ((A ↑m B) = ∅ ↔ {f ∣ f:B–→A} = ∅)
5		snssi 3852	. . . . . . . 8 ⊢ (x ∈ A → {x} ⊆ A)
6		vex 2862	. . . . . . . . . 10 ⊢ x ∈ V
7	6	fconst 5250	. . . . . . . . 9 ⊢ (B × {x}):B–→{x}
8		fss 5230	. . . . . . . . 9 ⊢ (((B × {x}):B–→{x} ∧ {x} ⊆ A) → (B × {x}):B–→A)
9	7, 8	mpan 651	. . . . . . . 8 ⊢ ({x} ⊆ A → (B × {x}):B–→A)
10		snex 4111	. . . . . . . . . 10 ⊢ {x} ∈ V
11	2, 10	xpex 5115	. . . . . . . . 9 ⊢ (B × {x}) ∈ V
12		feq1 5210	. . . . . . . . 9 ⊢ (f = (B × {x}) → (f:B–→A ↔ (B × {x}):B–→A))
13	11, 12	spcev 2946	. . . . . . . 8 ⊢ ((B × {x}):B–→A → ∃f f:B–→A)
14	5, 9, 13	3syl 18	. . . . . . 7 ⊢ (x ∈ A → ∃f f:B–→A)
15	14	exlimiv 1634	. . . . . 6 ⊢ (∃x x ∈ A → ∃f f:B–→A)
16		n0 3559	. . . . . 6 ⊢ (A ≠ ∅ ↔ ∃x x ∈ A)
17		abn0 3568	. . . . . 6 ⊢ ({f ∣ f:B–→A} ≠ ∅ ↔ ∃f f:B–→A)
18	15, 16, 17	3imtr4i 257	. . . . 5 ⊢ (A ≠ ∅ → {f ∣ f:B–→A} ≠ ∅)
19	18	necon4i 2576	. . . 4 ⊢ ({f ∣ f:B–→A} = ∅ → A = ∅)
20	4, 19	sylbi 187	. . 3 ⊢ ((A ↑m B) = ∅ → A = ∅)
21	1	map0e 6023	. . . . . 6 ⊢ (A ↑m ∅) = {∅}
22		0ex 4110	. . . . . . . 8 ⊢ ∅ ∈ V
23	22	snid 3760	. . . . . . 7 ⊢ ∅ ∈ {∅}
24		ne0i 3556	. . . . . . 7 ⊢ (∅ ∈ {∅} → {∅} ≠ ∅)
25	23, 24	ax-mp 8	. . . . . 6 ⊢ {∅} ≠ ∅
26	21, 25	eqnetri 2533	. . . . 5 ⊢ (A ↑m ∅) ≠ ∅
27		oveq2 5531	. . . . . 6 ⊢ (B = ∅ → (A ↑m B) = (A ↑m ∅))
28	27	neeq1d 2529	. . . . 5 ⊢ (B = ∅ → ((A ↑m B) ≠ ∅ ↔ (A ↑m ∅) ≠ ∅))
29	26, 28	mpbiri 224	. . . 4 ⊢ (B = ∅ → (A ↑m B) ≠ ∅)
30	29	necon2i 2563	. . 3 ⊢ ((A ↑m B) = ∅ → B ≠ ∅)
31	20, 30	jca 518	. 2 ⊢ ((A ↑m B) = ∅ → (A = ∅ ∧ B ≠ ∅))
32		oveq1 5530	. . 3 ⊢ (A = ∅ → (A ↑m B) = (∅ ↑m B))
33	2	map0b 6024	. . 3 ⊢ (B ≠ ∅ → (∅ ↑m B) = ∅)
34	32, 33	sylan9eq 2405	. 2 ⊢ ((A = ∅ ∧ B ≠ ∅) → (A ↑m B) = ∅)
35	31, 34	impbii 180	1 ⊢ ((A ↑m B) = ∅ ↔ (A = ∅ ∧ B ≠ ∅))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339   ≠ wne 2516  Vcvv 2859   ⊆ wss 3257  ∅c0 3550  {csn 3737   × cxp 4770  –→wf 4777  (class class class)co 5525   ↑_m cmap 5999

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt2 5654  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-map 6001

This theorem is referenced by: (None)