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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 5700 | . . 3 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V)) | |
2 | inxp 5844 | . . 3 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) | |
3 | inv1 4403 | . . . 4 ⊢ (𝐵 ∩ V) = 𝐵 | |
4 | 3 | xpeq2i 5715 | . . 3 ⊢ ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ((𝐴 ∩ 𝐶) × 𝐵) |
5 | 1, 2, 4 | 3eqtri 2766 | . 2 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 ∩ 𝐶) × 𝐵) |
6 | sseqin2 4230 | . . . 4 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 𝐶) | |
7 | 6 | biimpi 216 | . . 3 ⊢ (𝐶 ⊆ 𝐴 → (𝐴 ∩ 𝐶) = 𝐶) |
8 | 7 | xpeq1d 5717 | . 2 ⊢ (𝐶 ⊆ 𝐴 → ((𝐴 ∩ 𝐶) × 𝐵) = (𝐶 × 𝐵)) |
9 | 5, 8 | eqtrid 2786 | 1 ⊢ (𝐶 ⊆ 𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 Vcvv 3477 ∩ cin 3961 ⊆ wss 3962 × cxp 5686 ↾ cres 5690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-opab 5210 df-xp 5694 df-rel 5695 df-res 5700 |
This theorem is referenced by: fparlem3 8137 fparlem4 8138 fpwwe2lem12 10679 pwssplit3 21077 cnconst2 23306 xkoccn 23642 tmdgsum 24118 dvcmul 25995 dvcmulf 25996 ply1gsumz 33598 lbsdiflsp0 33653 dvsconst 44325 dvsid 44326 aacllem 49031 |
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