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| Mirrors > Home > MPE Home > Th. List > mapsspw | Structured version Visualization version GIF version | ||
| Description: Set exponentiation is a subset of the power set of the Cartesian product of its arguments. (Contributed by NM, 8-Dec-2006.) (Revised by Mario Carneiro, 26-Apr-2015.) |
| Ref | Expression |
|---|---|
| mapsspw | ⊢ (𝐴 ↑m 𝐵) ⊆ 𝒫 (𝐵 × 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mapsspm 8916 | . 2 ⊢ (𝐴 ↑m 𝐵) ⊆ (𝐴 ↑pm 𝐵) | |
| 2 | pmsspw 8917 | . 2 ⊢ (𝐴 ↑pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴) | |
| 3 | 1, 2 | sstri 3993 | 1 ⊢ (𝐴 ↑m 𝐵) ⊆ 𝒫 (𝐵 × 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ⊆ wss 3951 𝒫 cpw 4600 × cxp 5683 (class class class)co 7431 ↑m cmap 8866 ↑pm cpm 8867 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-pm 8869 |
| This theorem is referenced by: mapfi 9388 rankmapu 9918 grumap 10848 wunfunc 17946 |
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