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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 33866 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 33866. (Contributed by Scott Fenton, 31-Mar-2012.) |
Ref | Expression |
---|---|
xrnss3v | ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xrn 36187 | . 2 ⊢ (𝐴 ⋉ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) | |
2 | inss1 4129 | . . 3 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (◡(1st ↾ (V × V)) ∘ 𝐴) | |
3 | relco 6088 | . . . 4 ⊢ Rel (◡(1st ↾ (V × V)) ∘ 𝐴) | |
4 | vex 3402 | . . . . . . . . 9 ⊢ 𝑧 ∈ V | |
5 | vex 3402 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
6 | 4, 5 | brcnv 5736 | . . . . . . . 8 ⊢ (𝑧◡(1st ↾ (V × V))𝑦 ↔ 𝑦(1st ↾ (V × V))𝑧) |
7 | 4 | brresi 5845 | . . . . . . . . 9 ⊢ (𝑦(1st ↾ (V × V))𝑧 ↔ (𝑦 ∈ (V × V) ∧ 𝑦1st 𝑧)) |
8 | 7 | simplbi 501 | . . . . . . . 8 ⊢ (𝑦(1st ↾ (V × V))𝑧 → 𝑦 ∈ (V × V)) |
9 | 6, 8 | sylbi 220 | . . . . . . 7 ⊢ (𝑧◡(1st ↾ (V × V))𝑦 → 𝑦 ∈ (V × V)) |
10 | 9 | adantl 485 | . . . . . 6 ⊢ ((𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V)) |
11 | 10 | exlimiv 1938 | . . . . 5 ⊢ (∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V)) |
12 | vex 3402 | . . . . . 6 ⊢ 𝑥 ∈ V | |
13 | 12, 5 | opelco 5725 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (◡(1st ↾ (V × V)) ∘ 𝐴) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦)) |
14 | opelxp 5572 | . . . . . 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 295 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (◡(1st ↾ (V × V)) ∘ 𝐴) → 〈𝑥, 𝑦〉 ∈ (V × (V × V))) |
17 | 3, 16 | relssi 5642 | . . 3 ⊢ (◡(1st ↾ (V × V)) ∘ 𝐴) ⊆ (V × (V × V)) |
18 | 2, 17 | sstri 3896 | . 2 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (V × (V × V)) |
19 | 1, 18 | eqsstri 3921 | 1 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∃wex 1787 ∈ wcel 2112 Vcvv 3398 ∩ cin 3852 ⊆ wss 3853 〈cop 4533 class class class wbr 5039 × cxp 5534 ◡ccnv 5535 ↾ cres 5538 ∘ ccom 5540 1st c1st 7737 2nd c2nd 7738 ⋉ cxrn 36018 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-res 5548 df-xrn 36187 |
This theorem is referenced by: xrnrel 36189 brxrn2 36191 |
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