Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 6647 | . . . 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 6648 | . . 3 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 𝐵) → 𝑓:𝐵⟶𝐴) | |
5 | fpmg 6652 | . . 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 5194 (class class class)co 5853 ↑𝑚 cmap 6626 ↑pm cpm 6627 |
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 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 |
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 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-map 6628 df-pm 6629 |
This theorem is referenced by: mapsspw 6662 |
Copyright terms: Public domain | W3C validator |