Theorem map0b 6024

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 4110	. . 3 ⊢ ∅ ∈ V
2		map0e.1	. . 3 ⊢ A ∈ V
3	1, 2	mapval 6011	. 2 ⊢ (∅ ↑m A) = {f ∣ f:A–→∅}
4		abn0 3568	. . . 4 ⊢ ({f ∣ f:A–→∅} ≠ ∅ ↔ ∃f f:A–→∅)
5		fdm 5226	. . . . . 6 ⊢ (f:A–→∅ → dom f = A)
6		frn 5228	. . . . . . . 8 ⊢ (f:A–→∅ → ran f ⊆ ∅)
7		ss0 3581	. . . . . . . 8 ⊢ (ran f ⊆ ∅ → ran f = ∅)
8	6, 7	syl 15	. . . . . . 7 ⊢ (f:A–→∅ → ran f = ∅)
9		dm0rn0 4921	. . . . . . 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 2560	. 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 2516  Vcvv 2859   ⊆ wss 3257  ∅c0 3550  dom cdm 4772  ran crn 4773  –→wf 4777  (class class class)co 5525   ↑_m cmap 5999

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 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:  map0  6025