![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5531 | . . 3 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 × 𝐵) ∩ (𝐶 × V)) | |
2 | inxp 5667 | . . 3 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) | |
3 | inv1 4302 | . . . 4 ⊢ (𝐵 ∩ V) = 𝐵 | |
4 | 3 | xpeq2i 5546 | . . 3 ⊢ ((𝐴 ∩ 𝐶) × (𝐵 ∩ V)) = ((𝐴 ∩ 𝐶) × 𝐵) |
5 | 1, 2, 4 | 3eqtri 2825 | . 2 ⊢ ((𝐴 × 𝐵) ↾ 𝐶) = ((𝐴 ∩ 𝐶) × 𝐵) |
6 | sseqin2 4142 | . . . 4 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 𝐶) | |
7 | 6 | biimpi 219 | . . 3 ⊢ (𝐶 ⊆ 𝐴 → (𝐴 ∩ 𝐶) = 𝐶) |
8 | 7 | xpeq1d 5548 | . 2 ⊢ (𝐶 ⊆ 𝐴 → ((𝐴 ∩ 𝐶) × 𝐵) = (𝐶 × 𝐵)) |
9 | 5, 8 | syl5eq 2845 | 1 ⊢ (𝐶 ⊆ 𝐴 → ((𝐴 × 𝐵) ↾ 𝐶) = (𝐶 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 × cxp 5517 ↾ cres 5521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-opab 5093 df-xp 5525 df-rel 5526 df-res 5531 |
This theorem is referenced by: fparlem3 7792 fparlem4 7793 fpwwe2lem13 10053 pwssplit3 19826 cnconst2 21888 xkoccn 22224 tmdgsum 22700 dvcmul 24547 dvcmulf 24548 lbsdiflsp0 31110 dvsconst 41034 dvsid 41035 aacllem 45329 |
Copyright terms: Public domain | W3C validator |