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| Mirrors > Home > MPE Home > Th. List > xpss2 | Structured version Visualization version GIF version | ||
| Description: Subset relation for Cartesian product. (Contributed by Jeff Hankins, 30-Aug-2009.) |
| Ref | Expression |
|---|---|
| xpss2 | ⊢ (𝐴 ⊆ 𝐵 → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssid 4006 | . 2 ⊢ 𝐶 ⊆ 𝐶 | |
| 2 | xpss12 5700 | . 2 ⊢ ((𝐶 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐵) → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵)) | |
| 3 | 1, 2 | mpan 690 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ⊆ wss 3951 × cxp 5683 |
| 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-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ss 3968 df-opab 5206 df-xp 5691 |
| This theorem is referenced by: xpdom3 9110 marypha1lem 9473 canthp1lem2 10693 axresscn 11188 imasvscafn 17582 imasvscaf 17584 gass 19319 gsum2d 19990 pzriprnglem4 21495 pzriprnglem10 21501 tx2cn 23618 txtube 23648 txcmplem1 23649 hausdiag 23653 xkoinjcn 23695 caussi 25331 dvfval 25932 issh2 31228 elrgspnsubrunlem2 33252 qtophaus 33835 2ndmbfm 34263 sxbrsigalem0 34273 cvmlift2lem9 35316 cvmlift2lem11 35318 filnetlem3 36381 bj-idres 37161 idresssidinxp 38309 trclexi 43633 cnvtrcl0 43639 ovolval5lem2 46668 ovnovollem1 46671 ovnovollem2 46672 |
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