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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 8888 | . . . 4 ⊢ (𝑓 ∈ (∅ ↑m 𝐴) → 𝑓:𝐴⟶∅) | |
2 | fdm 6746 | . . . . 5 ⊢ (𝑓:𝐴⟶∅ → dom 𝑓 = 𝐴) | |
3 | frn 6744 | . . . . . . 7 ⊢ (𝑓:𝐴⟶∅ → ran 𝑓 ⊆ ∅) | |
4 | ss0 4408 | . . . . . . 7 ⊢ (ran 𝑓 ⊆ ∅ → ran 𝑓 = ∅) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (𝑓:𝐴⟶∅ → ran 𝑓 = ∅) |
6 | dm0rn0 5938 | . . . . . 6 ⊢ (dom 𝑓 = ∅ ↔ ran 𝑓 = ∅) | |
7 | 5, 6 | sylibr 234 | . . . . 5 ⊢ (𝑓:𝐴⟶∅ → dom 𝑓 = ∅) |
8 | 2, 7 | eqtr3d 2777 | . . . 4 ⊢ (𝑓:𝐴⟶∅ → 𝐴 = ∅) |
9 | 1, 8 | syl 17 | . . 3 ⊢ (𝑓 ∈ (∅ ↑m 𝐴) → 𝐴 = ∅) |
10 | 9 | necon3ai 2963 | . 2 ⊢ (𝐴 ≠ ∅ → ¬ 𝑓 ∈ (∅ ↑m 𝐴)) |
11 | 10 | eq0rdv 4413 | 1 ⊢ (𝐴 ≠ ∅ → (∅ ↑m 𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 ∅c0 4339 dom cdm 5689 ran crn 5690 ⟶wf 6559 (class class class)co 7431 ↑m cmap 8865 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 |
This theorem is referenced by: map0g 8923 mapdom2 9187 ply1plusgfvi 22259 satf0 35357 prv0 35415 |
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