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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 8637 | . . . 4 ⊢ (𝑓 ∈ (∅ ↑m 𝐴) → 𝑓:𝐴⟶∅) | |
2 | fdm 6609 | . . . . 5 ⊢ (𝑓:𝐴⟶∅ → dom 𝑓 = 𝐴) | |
3 | frn 6607 | . . . . . . 7 ⊢ (𝑓:𝐴⟶∅ → ran 𝑓 ⊆ ∅) | |
4 | ss0 4332 | . . . . . . 7 ⊢ (ran 𝑓 ⊆ ∅ → ran 𝑓 = ∅) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝑓:𝐴⟶∅ → ran 𝑓 = ∅) |
6 | dm0rn0 5834 | . . . . . 6 ⊢ (dom 𝑓 = ∅ ↔ ran 𝑓 = ∅) | |
7 | 5, 6 | sylibr 233 | . . . . 5 ⊢ (𝑓:𝐴⟶∅ → dom 𝑓 = ∅) |
8 | 2, 7 | eqtr3d 2780 | . . . 4 ⊢ (𝑓:𝐴⟶∅ → 𝐴 = ∅) |
9 | 1, 8 | syl 17 | . . 3 ⊢ (𝑓 ∈ (∅ ↑m 𝐴) → 𝐴 = ∅) |
10 | 9 | necon3ai 2968 | . 2 ⊢ (𝐴 ≠ ∅ → ¬ 𝑓 ∈ (∅ ↑m 𝐴)) |
11 | 10 | eq0rdv 4338 | 1 ⊢ (𝐴 ≠ ∅ → (∅ ↑m 𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ⊆ wss 3887 ∅c0 4256 dom cdm 5589 ran crn 5590 ⟶wf 6429 (class class class)co 7275 ↑m cmap 8615 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-map 8617 |
This theorem is referenced by: map0g 8672 mapdom2 8935 ply1plusgfvi 21413 satf0 33334 prv0 33392 |
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