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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 8789 | . . . 4 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
2 | 1 | simprd 497 | . . 3 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) → 𝐵 ∈ V) |
3 | 1 | simpld 496 | . . 3 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) → 𝐴 ∈ V) |
4 | elmapi 8790 | . . 3 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) → 𝑓:𝐵⟶𝐴) | |
5 | fpmg 8809 | . . 3 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V ∧ 𝑓:𝐵⟶𝐴) → 𝑓 ∈ (𝐴 ↑pm 𝐵)) | |
6 | 2, 3, 4, 5 | syl3anc 1372 | . 2 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) → 𝑓 ∈ (𝐴 ↑pm 𝐵)) |
7 | 6 | ssriv 3949 | 1 ⊢ (𝐴 ↑m 𝐵) ⊆ (𝐴 ↑pm 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 Vcvv 3444 ⊆ wss 3911 ⟶wf 6493 (class class class)co 7358 ↑m cmap 8768 ↑pm cpm 8769 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-map 8770 df-pm 8771 |
This theorem is referenced by: mapsspw 8819 wunmap 10667 dvntaylp 25746 taylthlem1 25748 taylthlem2 25749 mrsubrn 34164 mrsubff1 34165 msubrn 34180 msubff1 34207 |
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