Theorem map0b 6958

Description: Set exponentiation with an empty base is the empty set, provided the exponent is nonempty. Theorem 96 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)

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		elmapi 6934	. . . 4 ⊢ (𝑓 ∈ (∅ ↑𝑚 𝐴) → 𝑓:𝐴⟶∅)
2		fdm 5534	. . . . 5 ⊢ (𝑓:𝐴⟶∅ → dom 𝑓 = 𝐴)
3		frn 5537	. . . . . . 7 ⊢ (𝑓:𝐴⟶∅ → ran 𝑓 ⊆ ∅)
4		ss0 3563	. . . . . . 7 ⊢ (ran 𝑓 ⊆ ∅ → ran 𝑓 = ∅)
5	3, 4	syl 14	. . . . . 6 ⊢ (𝑓:𝐴⟶∅ → ran 𝑓 = ∅)
6		dm0rn0 4993	. . . . . 6 ⊢ (dom 𝑓 = ∅ ↔ ran 𝑓 = ∅)
7	5, 6	sylibr 134	. . . . 5 ⊢ (𝑓:𝐴⟶∅ → dom 𝑓 = ∅)
8	2, 7	eqtr3d 2273	. . . 4 ⊢ (𝑓:𝐴⟶∅ → 𝐴 = ∅)
9	1, 8	syl 14	. . 3 ⊢ (𝑓 ∈ (∅ ↑𝑚 𝐴) → 𝐴 = ∅)
10	9	necon3ai 2469	. 2 ⊢ (𝐴 ≠ ∅ → ¬ 𝑓 ∈ (∅ ↑𝑚 𝐴))
11	10	eq0rdv 3570	1 ⊢ (𝐴 ≠ ∅ → (∅ ↑𝑚 𝐴) = ∅)

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209   ≠ wne 2420   ⊆ wss 3220  ∅c0 3520  dom cdm 4769  ran crn 4770  ⟶wf 5368  (class class class)co 6075   ↑_𝑚 cmap 6912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-map 6914

This theorem is referenced by:  map0g  6959