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Mirrors > Home > MPE Home > Th. List > Mathboxes > txpss3v | Structured version Visualization version GIF version |
Description: A tail Cartesian product is a subset of the class of ordered triples. (Contributed by Scott Fenton, 31-Mar-2012.) |
Ref | Expression |
---|---|
txpss3v | ⊢ (𝐴 ⊗ 𝐵) ⊆ (V × (V × V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-txp 33310 | . 2 ⊢ (𝐴 ⊗ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) | |
2 | inss1 4204 | . . 3 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (◡(1st ↾ (V × V)) ∘ 𝐴) | |
3 | relco 6091 | . . . 4 ⊢ Rel (◡(1st ↾ (V × V)) ∘ 𝐴) | |
4 | vex 3497 | . . . . . . . . 9 ⊢ 𝑧 ∈ V | |
5 | vex 3497 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
6 | 4, 5 | brcnv 5747 | . . . . . . . 8 ⊢ (𝑧◡(1st ↾ (V × V))𝑦 ↔ 𝑦(1st ↾ (V × V))𝑧) |
7 | 4 | brresi 5856 | . . . . . . . . 9 ⊢ (𝑦(1st ↾ (V × V))𝑧 ↔ (𝑦 ∈ (V × V) ∧ 𝑦1st 𝑧)) |
8 | 7 | simplbi 500 | . . . . . . . 8 ⊢ (𝑦(1st ↾ (V × V))𝑧 → 𝑦 ∈ (V × V)) |
9 | 6, 8 | sylbi 219 | . . . . . . 7 ⊢ (𝑧◡(1st ↾ (V × V))𝑦 → 𝑦 ∈ (V × V)) |
10 | 9 | adantl 484 | . . . . . 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 3497 | . . . . . 6 ⊢ 𝑥 ∈ V | |
13 | 12, 5 | opelco 5736 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (◡(1st ↾ (V × V)) ∘ 𝐴) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦)) |
14 | opelxp 5585 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ (V × (V × V)) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ (V × V))) | |
15 | 12, 14 | mpbiran 707 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (V × (V × V)) ↔ 𝑦 ∈ (V × V)) |
16 | 11, 13, 15 | 3imtr4i 294 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (◡(1st ↾ (V × V)) ∘ 𝐴) → 〈𝑥, 𝑦〉 ∈ (V × (V × V))) |
17 | 3, 16 | relssi 5654 | . . 3 ⊢ (◡(1st ↾ (V × V)) ∘ 𝐴) ⊆ (V × (V × V)) |
18 | 2, 17 | sstri 3975 | . 2 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (V × (V × V)) |
19 | 1, 18 | eqsstri 4000 | 1 ⊢ (𝐴 ⊗ 𝐵) ⊆ (V × (V × V)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∃wex 1776 ∈ wcel 2110 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 〈cop 4566 class class class wbr 5058 × cxp 5547 ◡ccnv 5548 ↾ cres 5551 ∘ ccom 5553 1st c1st 7681 2nd c2nd 7682 ⊗ ctxp 33286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-res 5561 df-txp 33310 |
This theorem is referenced by: txprel 33335 brtxp2 33337 pprodss4v 33340 |
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