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Mirrors > Home > MPE Home > Th. List > mapsspm | Structured version Visualization version GIF version |
Description: Set exponentiation is a subset of partial maps. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 27-Feb-2016.) |
Ref | Expression |
---|---|
mapsspm | ⊢ (𝐴 ↑m 𝐵) ⊆ (𝐴 ↑pm 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapex 8841 | . . . 4 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
2 | 1 | simprd 495 | . . 3 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) → 𝐵 ∈ V) |
3 | 1 | simpld 494 | . . 3 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) → 𝐴 ∈ V) |
4 | elmapi 8842 | . . 3 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) → 𝑓:𝐵⟶𝐴) | |
5 | fpmg 8861 | . . 3 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V ∧ 𝑓:𝐵⟶𝐴) → 𝑓 ∈ (𝐴 ↑pm 𝐵)) | |
6 | 2, 3, 4, 5 | syl3anc 1368 | . 2 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) → 𝑓 ∈ (𝐴 ↑pm 𝐵)) |
7 | 6 | ssriv 3981 | 1 ⊢ (𝐴 ↑m 𝐵) ⊆ (𝐴 ↑pm 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 Vcvv 3468 ⊆ wss 3943 ⟶wf 6532 (class class class)co 7404 ↑m cmap 8819 ↑pm cpm 8820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-map 8821 df-pm 8822 |
This theorem is referenced by: mapsspw 8871 wunmap 10720 dvntaylp 26257 taylthlem1 26259 taylthlem2 26260 taylthlem2OLD 26261 mrsubrn 35032 mrsubff1 35033 msubrn 35048 msubff1 35075 |
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