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| Mirrors > Home > MPE Home > Th. List > xpssres | Structured version Visualization version GIF version | ||
| Description: Restriction of a constant function (or other Cartesian product). (Contributed by Stefan O'Rear, 24-Jan-2015.) |
| Ref | Expression |
|---|---|
| xpssres | ⊢ (𝐶 ⊆ 𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-res 5697 | . . 3 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V)) | |
| 2 | inxp 5842 | . . 3 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) | |
| 3 | inv1 4398 | . . . 4 ⊢ (𝐵 ∩ V) = 𝐵 | |
| 4 | 3 | xpeq2i 5712 | . . 3 ⊢ ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ((𝐴 ∩ 𝐶) × 𝐵) |
| 5 | 1, 2, 4 | 3eqtri 2769 | . 2 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 ∩ 𝐶) × 𝐵) |
| 6 | sseqin2 4223 | . . . 4 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 𝐶) | |
| 7 | 6 | biimpi 216 | . . 3 ⊢ (𝐶 ⊆ 𝐴 → (𝐴 ∩ 𝐶) = 𝐶) |
| 8 | 7 | xpeq1d 5714 | . 2 ⊢ (𝐶 ⊆ 𝐴 → ((𝐴 ∩ 𝐶) × 𝐵) = (𝐶 × 𝐵)) |
| 9 | 5, 8 | eqtrid 2789 | 1 ⊢ (𝐶 ⊆ 𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 × cxp 5683 ↾ cres 5687 |
| 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 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 df-xp 5691 df-rel 5692 df-res 5697 |
| This theorem is referenced by: fparlem3 8139 fparlem4 8140 fpwwe2lem12 10682 pwssplit3 21060 cnconst2 23291 xkoccn 23627 tmdgsum 24103 dvcmul 25981 dvcmulf 25982 ply1gsumz 33619 lbsdiflsp0 33677 dvsconst 44349 dvsid 44350 aacllem 49320 |
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