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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 8889 | . . . 4 ⊢ (𝑓 ∈ (∅ ↑m 𝐴) → 𝑓:𝐴⟶∅) | |
| 2 | fdm 6745 | . . . . 5 ⊢ (𝑓:𝐴⟶∅ → dom 𝑓 = 𝐴) | |
| 3 | frn 6743 | . . . . . . 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 5935 | . . . . . 6 ⊢ (dom 𝑓 = ∅ ↔ ran 𝑓 = ∅) | |
| 7 | 5, 6 | sylibr 234 | . . . . 5 ⊢ (𝑓:𝐴⟶∅ → dom 𝑓 = ∅) |
| 8 | 2, 7 | eqtr3d 2779 | . . . 4 ⊢ (𝑓:𝐴⟶∅ → 𝐴 = ∅) |
| 9 | 1, 8 | syl 17 | . . 3 ⊢ (𝑓 ∈ (∅ ↑m 𝐴) → 𝐴 = ∅) |
| 10 | 9 | necon3ai 2965 | . 2 ⊢ (𝐴 ≠ ∅ → ¬ 𝑓 ∈ (∅ ↑m 𝐴)) |
| 11 | 10 | eq0rdv 4407 | 1 ⊢ (𝐴 ≠ ∅ → (∅ ↑m 𝐴) = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 ∅c0 4333 dom cdm 5685 ran crn 5686 ⟶wf 6557 (class class class)co 7431 ↑m cmap 8866 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 |
| This theorem is referenced by: map0g 8924 mapdom2 9188 ply1plusgfvi 22243 satf0 35377 prv0 35435 |
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