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Mirrors > Home > ILE Home > Th. List > map0b | 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 6572 | . . . 4 ⊢ (𝑓 ∈ (∅ ↑𝑚 𝐴) → 𝑓:𝐴⟶∅) | |
2 | fdm 5286 | . . . . 5 ⊢ (𝑓:𝐴⟶∅ → dom 𝑓 = 𝐴) | |
3 | frn 5289 | . . . . . . 7 ⊢ (𝑓:𝐴⟶∅ → ran 𝑓 ⊆ ∅) | |
4 | ss0 3408 | . . . . . . 7 ⊢ (ran 𝑓 ⊆ ∅ → ran 𝑓 = ∅) | |
5 | 3, 4 | syl 14 | . . . . . 6 ⊢ (𝑓:𝐴⟶∅ → ran 𝑓 = ∅) |
6 | dm0rn0 4764 | . . . . . 6 ⊢ (dom 𝑓 = ∅ ↔ ran 𝑓 = ∅) | |
7 | 5, 6 | sylibr 133 | . . . . 5 ⊢ (𝑓:𝐴⟶∅ → dom 𝑓 = ∅) |
8 | 2, 7 | eqtr3d 2175 | . . . 4 ⊢ (𝑓:𝐴⟶∅ → 𝐴 = ∅) |
9 | 1, 8 | syl 14 | . . 3 ⊢ (𝑓 ∈ (∅ ↑𝑚 𝐴) → 𝐴 = ∅) |
10 | 9 | necon3ai 2358 | . 2 ⊢ (𝐴 ≠ ∅ → ¬ 𝑓 ∈ (∅ ↑𝑚 𝐴)) |
11 | 10 | eq0rdv 3412 | 1 ⊢ (𝐴 ≠ ∅ → (∅ ↑𝑚 𝐴) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 ≠ wne 2309 ⊆ wss 3076 ∅c0 3368 dom cdm 4547 ran crn 4548 ⟶wf 5127 (class class class)co 5782 ↑𝑚 cmap 6550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-map 6552 |
This theorem is referenced by: map0g 6590 |
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