Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 8421 | . . . 4 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
2 | 1 | simprd 498 | . . 3 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) → 𝐵 ∈ V) |
3 | 1 | simpld 497 | . . 3 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) → 𝐴 ∈ V) |
4 | elmapi 8422 | . . 3 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) → 𝑓:𝐵⟶𝐴) | |
5 | fpmg 8426 | . . 3 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V ∧ 𝑓:𝐵⟶𝐴) → 𝑓 ∈ (𝐴 ↑pm 𝐵)) | |
6 | 2, 3, 4, 5 | syl3anc 1367 | . 2 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) → 𝑓 ∈ (𝐴 ↑pm 𝐵)) |
7 | 6 | ssriv 3971 | 1 ⊢ (𝐴 ↑m 𝐵) ⊆ (𝐴 ↑pm 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 Vcvv 3495 ⊆ wss 3936 ⟶wf 6346 (class class class)co 7150 ↑m cmap 8400 ↑pm cpm 8401 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-map 8402 df-pm 8403 |
This theorem is referenced by: mapsspw 8436 wunmap 10142 dvntaylp 24953 taylthlem1 24955 taylthlem2 24956 mrsubrn 32755 mrsubff1 32756 msubrn 32771 msubff1 32798 |
Copyright terms: Public domain | W3C validator |