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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 8422 | . . . 4 ⊢ (𝑓 ∈ (∅ ↑m 𝐴) → 𝑓:𝐴⟶∅) | |
2 | fdm 6516 | . . . . 5 ⊢ (𝑓:𝐴⟶∅ → dom 𝑓 = 𝐴) | |
3 | frn 6514 | . . . . . . 7 ⊢ (𝑓:𝐴⟶∅ → ran 𝑓 ⊆ ∅) | |
4 | ss0 4351 | . . . . . . 7 ⊢ (ran 𝑓 ⊆ ∅ → ran 𝑓 = ∅) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝑓:𝐴⟶∅ → ran 𝑓 = ∅) |
6 | dm0rn0 5789 | . . . . . 6 ⊢ (dom 𝑓 = ∅ ↔ ran 𝑓 = ∅) | |
7 | 5, 6 | sylibr 236 | . . . . 5 ⊢ (𝑓:𝐴⟶∅ → dom 𝑓 = ∅) |
8 | 2, 7 | eqtr3d 2858 | . . . 4 ⊢ (𝑓:𝐴⟶∅ → 𝐴 = ∅) |
9 | 1, 8 | syl 17 | . . 3 ⊢ (𝑓 ∈ (∅ ↑m 𝐴) → 𝐴 = ∅) |
10 | 9 | necon3ai 3041 | . 2 ⊢ (𝐴 ≠ ∅ → ¬ 𝑓 ∈ (∅ ↑m 𝐴)) |
11 | 10 | eq0rdv 4356 | 1 ⊢ (𝐴 ≠ ∅ → (∅ ↑m 𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ⊆ wss 3935 ∅c0 4290 dom cdm 5549 ran crn 5550 ⟶wf 6345 (class class class)co 7150 ↑m cmap 8400 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-map 8402 |
This theorem is referenced by: map0g 8442 mapdom2 8682 ply1plusgfvi 20404 satf0 32614 prv0 32672 |
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