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Mirrors > Home > MPE Home > Th. List > map0b | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
map0b | ⊢ (𝐴 ≠ ∅ → (∅ ↑m 𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapi 8874 | . . . 4 ⊢ (𝑓 ∈ (∅ ↑m 𝐴) → 𝑓:𝐴⟶∅) | |
2 | fdm 6736 | . . . . 5 ⊢ (𝑓:𝐴⟶∅ → dom 𝑓 = 𝐴) | |
3 | frn 6734 | . . . . . . 7 ⊢ (𝑓:𝐴⟶∅ → ran 𝑓 ⊆ ∅) | |
4 | ss0 4402 | . . . . . . 7 ⊢ (ran 𝑓 ⊆ ∅ → ran 𝑓 = ∅) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝑓:𝐴⟶∅ → ran 𝑓 = ∅) |
6 | dm0rn0 5931 | . . . . . 6 ⊢ (dom 𝑓 = ∅ ↔ ran 𝑓 = ∅) | |
7 | 5, 6 | sylibr 233 | . . . . 5 ⊢ (𝑓:𝐴⟶∅ → dom 𝑓 = ∅) |
8 | 2, 7 | eqtr3d 2770 | . . . 4 ⊢ (𝑓:𝐴⟶∅ → 𝐴 = ∅) |
9 | 1, 8 | syl 17 | . . 3 ⊢ (𝑓 ∈ (∅ ↑m 𝐴) → 𝐴 = ∅) |
10 | 9 | necon3ai 2962 | . 2 ⊢ (𝐴 ≠ ∅ → ¬ 𝑓 ∈ (∅ ↑m 𝐴)) |
11 | 10 | eq0rdv 4408 | 1 ⊢ (𝐴 ≠ ∅ → (∅ ↑m 𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ⊆ wss 3949 ∅c0 4326 dom cdm 5682 ran crn 5683 ⟶wf 6549 (class class class)co 7426 ↑m cmap 8851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-map 8853 |
This theorem is referenced by: map0g 8909 mapdom2 9179 ply1plusgfvi 22167 satf0 35015 prv0 35073 |
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