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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrnss3v | Structured version Visualization version GIF version |
Description: A range Cartesian product is a subset of the class of ordered triples. This is Scott Fenton's txpss3v 35859 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 35859. (Contributed by Scott Fenton, 31-Mar-2012.) |
Ref | Expression |
---|---|
xrnss3v | ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xrn 38352 | . 2 ⊢ (𝐴 ⋉ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) | |
2 | inss1 4244 | . . 3 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (◡(1st ↾ (V × V)) ∘ 𝐴) | |
3 | relco 6128 | . . . 4 ⊢ Rel (◡(1st ↾ (V × V)) ∘ 𝐴) | |
4 | vex 3481 | . . . . . . . . 9 ⊢ 𝑧 ∈ V | |
5 | vex 3481 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
6 | 4, 5 | brcnv 5895 | . . . . . . . 8 ⊢ (𝑧◡(1st ↾ (V × V))𝑦 ↔ 𝑦(1st ↾ (V × V))𝑧) |
7 | 4 | brresi 6008 | . . . . . . . . 9 ⊢ (𝑦(1st ↾ (V × V))𝑧 ↔ (𝑦 ∈ (V × V) ∧ 𝑦1st 𝑧)) |
8 | 7 | simplbi 497 | . . . . . . . 8 ⊢ (𝑦(1st ↾ (V × V))𝑧 → 𝑦 ∈ (V × V)) |
9 | 6, 8 | sylbi 217 | . . . . . . 7 ⊢ (𝑧◡(1st ↾ (V × V))𝑦 → 𝑦 ∈ (V × V)) |
10 | 9 | adantl 481 | . . . . . 6 ⊢ ((𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V)) |
11 | 10 | exlimiv 1927 | . . . . 5 ⊢ (∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V)) |
12 | vex 3481 | . . . . . 6 ⊢ 𝑥 ∈ V | |
13 | 12, 5 | opelco 5884 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (◡(1st ↾ (V × V)) ∘ 𝐴) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦)) |
14 | opelxp 5724 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ (V × (V × V)) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ (V × V))) | |
15 | 12, 14 | mpbiran 709 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (V × (V × V)) ↔ 𝑦 ∈ (V × V)) |
16 | 11, 13, 15 | 3imtr4i 292 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (◡(1st ↾ (V × V)) ∘ 𝐴) → 〈𝑥, 𝑦〉 ∈ (V × (V × V))) |
17 | 3, 16 | relssi 5799 | . . 3 ⊢ (◡(1st ↾ (V × V)) ∘ 𝐴) ⊆ (V × (V × V)) |
18 | 2, 17 | sstri 4004 | . 2 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (V × (V × V)) |
19 | 1, 18 | eqsstri 4029 | 1 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∃wex 1775 ∈ wcel 2105 Vcvv 3477 ∩ cin 3961 ⊆ wss 3962 〈cop 4636 class class class wbr 5147 × cxp 5686 ◡ccnv 5687 ↾ cres 5690 ∘ ccom 5692 1st c1st 8010 2nd c2nd 8011 ⋉ cxrn 38160 |
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-br 5148 df-opab 5210 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-res 5700 df-xrn 38352 |
This theorem is referenced by: xrnrel 38354 brxrn2 38356 |
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