Theorem map0 6026

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 6012	. . . . 5 ⊢ (A ↑m B) = {f ∣ f:B–→A}
4	3	eqeq1i 2360	. . . 4 ⊢ ((A ↑m B) = ∅ ↔ {f ∣ f:B–→A} = ∅)
5		snssi 3853	. . . . . . . 8 ⊢ (x ∈ A → {x} ⊆ A)
6		vex 2863	. . . . . . . . . 10 ⊢ x ∈ V
7	6	fconst 5251	. . . . . . . . 9 ⊢ (B × {x}):B–→{x}
8		fss 5231	. . . . . . . . 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 4112	. . . . . . . . . 10 ⊢ {x} ∈ V
11	2, 10	xpex 5116	. . . . . . . . 9 ⊢ (B × {x}) ∈ V
12		feq1 5211	. . . . . . . . 9 ⊢ (f = (B × {x}) → (f:B–→A ↔ (B × {x}):B–→A))
13	11, 12	spcev 2947	. . . . . . . 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 3560	. . . . . 6 ⊢ (A ≠ ∅ ↔ ∃x x ∈ A)
17		abn0 3569	. . . . . 6 ⊢ ({f ∣ f:B–→A} ≠ ∅ ↔ ∃f f:B–→A)
18	15, 16, 17	3imtr4i 257	. . . . 5 ⊢ (A ≠ ∅ → {f ∣ f:B–→A} ≠ ∅)
19	18	necon4i 2577	. . . 4 ⊢ ({f ∣ f:B–→A} = ∅ → A = ∅)
20	4, 19	sylbi 187	. . 3 ⊢ ((A ↑m B) = ∅ → A = ∅)
21	1	map0e 6024	. . . . . 6 ⊢ (A ↑m ∅) = {∅}
22		0ex 4111	. . . . . . . 8 ⊢ ∅ ∈ V
23	22	snid 3761	. . . . . . 7 ⊢ ∅ ∈ {∅}
24		ne0i 3557	. . . . . . 7 ⊢ (∅ ∈ {∅} → {∅} ≠ ∅)
25	23, 24	ax-mp 5	. . . . . 6 ⊢ {∅} ≠ ∅
26	21, 25	eqnetri 2534	. . . . 5 ⊢ (A ↑m ∅) ≠ ∅
27		oveq2 5532	. . . . . 6 ⊢ (B = ∅ → (A ↑m B) = (A ↑m ∅))
28	27	neeq1d 2530	. . . . 5 ⊢ (B = ∅ → ((A ↑m B) ≠ ∅ ↔ (A ↑m ∅) ≠ ∅))
29	26, 28	mpbiri 224	. . . 4 ⊢ (B = ∅ → (A ↑m B) ≠ ∅)
30	29	necon2i 2564	. . 3 ⊢ ((A ↑m B) = ∅ → B ≠ ∅)
31	20, 30	jca 518	. 2 ⊢ ((A ↑m B) = ∅ → (A = ∅ ∧ B ≠ ∅))
32		oveq1 5531	. . 3 ⊢ (A = ∅ → (A ↑m B) = (∅ ↑m B))
33	2	map0b 6025	. . 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 ≠ ∅))

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339   ≠ wne 2517  Vcvv 2860   ⊆ wss 3258  ∅c0 3551  {csn 3738   × cxp 4771  –→wf 4778  (class class class)co 5526   ↑_m cmap 6000

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof 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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt2 5655  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-map 6002

This theorem is used by: (None)