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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 31130 | . 2 ⊢ (𝐴 ⊗ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) | |
2 | inss1 3795 | . . 3 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (◡(1st ↾ (V × V)) ∘ 𝐴) | |
3 | relco 5550 | . . . 4 ⊢ Rel (◡(1st ↾ (V × V)) ∘ 𝐴) | |
4 | vex 3176 | . . . . . . . . 9 ⊢ 𝑧 ∈ V | |
5 | vex 3176 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
6 | 4, 5 | brcnv 5227 | . . . . . . . 8 ⊢ (𝑧◡(1st ↾ (V × V))𝑦 ↔ 𝑦(1st ↾ (V × V))𝑧) |
7 | 4 | brres 5323 | . . . . . . . . 9 ⊢ (𝑦(1st ↾ (V × V))𝑧 ↔ (𝑦1st 𝑧 ∧ 𝑦 ∈ (V × V))) |
8 | 7 | simprbi 479 | . . . . . . . 8 ⊢ (𝑦(1st ↾ (V × V))𝑧 → 𝑦 ∈ (V × V)) |
9 | 6, 8 | sylbi 206 | . . . . . . 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 1845 | . . . . 5 ⊢ (∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V)) |
12 | vex 3176 | . . . . . 6 ⊢ 𝑥 ∈ V | |
13 | 12, 5 | opelco 5215 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (◡(1st ↾ (V × V)) ∘ 𝐴) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦)) |
14 | opelxp 5070 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ (V × (V × V)) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ (V × V))) | |
15 | 12, 14 | mpbiran 955 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (V × (V × V)) ↔ 𝑦 ∈ (V × V)) |
16 | 11, 13, 15 | 3imtr4i 280 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (◡(1st ↾ (V × V)) ∘ 𝐴) → 〈𝑥, 𝑦〉 ∈ (V × (V × V))) |
17 | 3, 16 | relssi 5134 | . . 3 ⊢ (◡(1st ↾ (V × V)) ∘ 𝐴) ⊆ (V × (V × V)) |
18 | 2, 17 | sstri 3577 | . 2 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (V × (V × V)) |
19 | 1, 18 | eqsstri 3598 | 1 ⊢ (𝐴 ⊗ 𝐵) ⊆ (V × (V × V)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 〈cop 4131 class class class wbr 4583 × cxp 5036 ◡ccnv 5037 ↾ cres 5040 ∘ ccom 5042 1st c1st 7057 2nd c2nd 7058 ⊗ ctxp 31106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-res 5050 df-txp 31130 |
This theorem is referenced by: txprel 31156 brtxp2 31158 pprodss4v 31161 |
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