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Mathbox for Scott Fenton |
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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 34895 | . 2 ⊢ (𝐴 ⊗ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) | |
2 | inss1 4228 | . . 3 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (◡(1st ↾ (V × V)) ∘ 𝐴) | |
3 | relco 6107 | . . . 4 ⊢ Rel (◡(1st ↾ (V × V)) ∘ 𝐴) | |
4 | vex 3478 | . . . . . . . . 9 ⊢ 𝑧 ∈ V | |
5 | vex 3478 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
6 | 4, 5 | brcnv 5882 | . . . . . . . 8 ⊢ (𝑧◡(1st ↾ (V × V))𝑦 ↔ 𝑦(1st ↾ (V × V))𝑧) |
7 | 4 | brresi 5990 | . . . . . . . . 9 ⊢ (𝑦(1st ↾ (V × V))𝑧 ↔ (𝑦 ∈ (V × V) ∧ 𝑦1st 𝑧)) |
8 | 7 | simplbi 498 | . . . . . . . 8 ⊢ (𝑦(1st ↾ (V × V))𝑧 → 𝑦 ∈ (V × V)) |
9 | 6, 8 | sylbi 216 | . . . . . . 7 ⊢ (𝑧◡(1st ↾ (V × V))𝑦 → 𝑦 ∈ (V × V)) |
10 | 9 | adantl 482 | . . . . . 6 ⊢ ((𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V)) |
11 | 10 | exlimiv 1933 | . . . . 5 ⊢ (∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V)) |
12 | vex 3478 | . . . . . 6 ⊢ 𝑥 ∈ V | |
13 | 12, 5 | opelco 5871 | . . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡(1st ↾ (V × V)) ∘ 𝐴) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦)) |
14 | opelxp 5712 | . . . . . 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 291 | . . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡(1st ↾ (V × V)) ∘ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (V × (V × V))) |
17 | 3, 16 | relssi 5787 | . . 3 ⊢ (◡(1st ↾ (V × V)) ∘ 𝐴) ⊆ (V × (V × V)) |
18 | 2, 17 | sstri 3991 | . 2 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (V × (V × V)) |
19 | 1, 18 | eqsstri 4016 | 1 ⊢ (𝐴 ⊗ 𝐵) ⊆ (V × (V × V)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∃wex 1781 ∈ wcel 2106 Vcvv 3474 ∩ cin 3947 ⊆ wss 3948 ⟨cop 4634 class class class wbr 5148 × cxp 5674 ◡ccnv 5675 ↾ cres 5678 ∘ ccom 5680 1st c1st 7975 2nd c2nd 7976 ⊗ ctxp 34871 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-res 5688 df-txp 34895 |
This theorem is referenced by: txprel 34920 brtxp2 34922 pprodss4v 34925 |
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