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Mirrors > Home > ILE Home > Th. List > mapss | GIF version |
Description: Subset inheritance for set exponentiation. Theorem 99 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
mapss | ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapi 6532 | . . . . . 6 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 𝐶) → 𝑓:𝐶⟶𝐴) | |
2 | 1 | adantl 275 | . . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑓 ∈ (𝐴 ↑𝑚 𝐶)) → 𝑓:𝐶⟶𝐴) |
3 | simplr 504 | . . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑓 ∈ (𝐴 ↑𝑚 𝐶)) → 𝐴 ⊆ 𝐵) | |
4 | 2, 3 | fssd 5255 | . . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑓 ∈ (𝐴 ↑𝑚 𝐶)) → 𝑓:𝐶⟶𝐵) |
5 | simpll 503 | . . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑓 ∈ (𝐴 ↑𝑚 𝐶)) → 𝐵 ∈ 𝑉) | |
6 | elmapex 6531 | . . . . . . 7 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 𝐶) → (𝐴 ∈ V ∧ 𝐶 ∈ V)) | |
7 | 6 | simprd 113 | . . . . . 6 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 𝐶) → 𝐶 ∈ V) |
8 | 7 | adantl 275 | . . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑓 ∈ (𝐴 ↑𝑚 𝐶)) → 𝐶 ∈ V) |
9 | 5, 8 | elmapd 6524 | . . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑓 ∈ (𝐴 ↑𝑚 𝐶)) → (𝑓 ∈ (𝐵 ↑𝑚 𝐶) ↔ 𝑓:𝐶⟶𝐵)) |
10 | 4, 9 | mpbird 166 | . . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) ∧ 𝑓 ∈ (𝐴 ↑𝑚 𝐶)) → 𝑓 ∈ (𝐵 ↑𝑚 𝐶)) |
11 | 10 | ex 114 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (𝑓 ∈ (𝐴 ↑𝑚 𝐶) → 𝑓 ∈ (𝐵 ↑𝑚 𝐶))) |
12 | 11 | ssrdv 3073 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1465 Vcvv 2660 ⊆ wss 3041 ⟶wf 5089 (class class class)co 5742 ↑𝑚 cmap 6510 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-map 6512 |
This theorem is referenced by: mapdom1g 6709 |
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