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Mirrors > Home > ILE Home > Th. List > mapsspm | 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 6643 | . . . 4 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
2 | 1 | simprd 113 | . . 3 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 𝐵) → 𝐵 ∈ V) |
3 | 1 | simpld 111 | . . 3 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 𝐵) → 𝐴 ∈ V) |
4 | elmapi 6644 | . . 3 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 𝐵) → 𝑓:𝐵⟶𝐴) | |
5 | fpmg 6648 | . . 3 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V ∧ 𝑓:𝐵⟶𝐴) → 𝑓 ∈ (𝐴 ↑pm 𝐵)) | |
6 | 2, 3, 4, 5 | syl3anc 1233 | . 2 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 𝐵) → 𝑓 ∈ (𝐴 ↑pm 𝐵)) |
7 | 6 | ssriv 3151 | 1 ⊢ (𝐴 ↑𝑚 𝐵) ⊆ (𝐴 ↑pm 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2141 Vcvv 2730 ⊆ wss 3121 ⟶wf 5192 (class class class)co 5850 ↑𝑚 cmap 6622 ↑pm cpm 6623 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 df-ov 5853 df-oprab 5854 df-mpo 5855 df-map 6624 df-pm 6625 |
This theorem is referenced by: mapsspw 6658 |
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