Theorem map0b 6025

Description: Set exponentiation with an empty base is the empty set, provided the exponent is nonempty. Theorem 96 of [Suppes] p. 89. (Contributed by set.mm contributors, 10-Dec-2003.) (Revised by set.mm contributors, 19-Mar-2007.)

Hypothesis

Ref	Expression
map0e.1	⊢ A ∈ V

Assertion

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

Proof of Theorem map0b

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4111	. . 3 ⊢ ∅ ∈ V
2		map0e.1	. . 3 ⊢ A ∈ V
3	1, 2	mapval 6012	. 2 ⊢ (∅ ↑m A) = {f ∣ f:A–→∅}
4		abn0 3569	. . . 4 ⊢ ({f ∣ f:A–→∅} ≠ ∅ ↔ ∃f f:A–→∅)
5		fdm 5227	. . . . . 6 ⊢ (f:A–→∅ → dom f = A)
6		frn 5229	. . . . . . . 8 ⊢ (f:A–→∅ → ran f ⊆ ∅)
7		ss0 3582	. . . . . . . 8 ⊢ (ran f ⊆ ∅ → ran f = ∅)
8	6, 7	syl 15	. . . . . . 7 ⊢ (f:A–→∅ → ran f = ∅)
9		dm0rn0 4922	. . . . . . 7 ⊢ (dom f = ∅ ↔ ran f = ∅)
10	8, 9	sylibr 203	. . . . . 6 ⊢ (f:A–→∅ → dom f = ∅)
11	5, 10	eqtr3d 2387	. . . . 5 ⊢ (f:A–→∅ → A = ∅)
12	11	exlimiv 1634	. . . 4 ⊢ (∃f f:A–→∅ → A = ∅)
13	4, 12	sylbi 187	. . 3 ⊢ ({f ∣ f:A–→∅} ≠ ∅ → A = ∅)
14	13	necon1i 2561	. 2 ⊢ (A ≠ ∅ → {f ∣ f:A–→∅} = ∅)
15	3, 14	syl5eq 2397	1 ⊢ (A ≠ ∅ → (∅ ↑m A) = ∅)

Syntax hints:   → wi 4  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339   ≠ wne 2517  Vcvv 2860   ⊆ wss 3258  ∅c0 3551  dom cdm 4773  ran crn 4774  –→wf 4778  (class class class)co 5526   ↑_m cmap 6000

This theorem was proved from 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 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 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 referenced by:  map0  6026