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Mirrors > Home > MPE Home > Th. List > xptrrel | Structured version Visualization version GIF version |
Description: The cross product is always a transitive relation. (Contributed by RP, 24-Dec-2019.) |
Ref | Expression |
---|---|
xptrrel | ⊢ ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 4155 | . . . . . . . 8 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ dom (𝐴 × 𝐵) | |
2 | dmxpss 5995 | . . . . . . . 8 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴 | |
3 | 1, 2 | sstri 3924 | . . . . . . 7 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ 𝐴 |
4 | inss2 4156 | . . . . . . . 8 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵) | |
5 | rnxpss 5996 | . . . . . . . 8 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 | |
6 | 4, 5 | sstri 3924 | . . . . . . 7 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ 𝐵 |
7 | 3, 6 | ssini 4158 | . . . . . 6 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ (𝐴 ∩ 𝐵) |
8 | eqimss 3971 | . . . . . 6 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∩ 𝐵) ⊆ ∅) | |
9 | 7, 8 | sstrid 3926 | . . . . 5 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ∅) |
10 | ss0 4306 | . . . . 5 ⊢ ((dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ∅ → (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) = ∅) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) = ∅) |
12 | 11 | coemptyd 14330 | . . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = ∅) |
13 | 0ss 4304 | . . 3 ⊢ ∅ ⊆ (𝐴 × 𝐵) | |
14 | 12, 13 | eqsstrdi 3969 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)) |
15 | neqne 2995 | . . . 4 ⊢ (¬ (𝐴 ∩ 𝐵) = ∅ → (𝐴 ∩ 𝐵) ≠ ∅) | |
16 | 15 | xpcoidgend 14326 | . . 3 ⊢ (¬ (𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵)) |
17 | ssid 3937 | . . 3 ⊢ (𝐴 × 𝐵) ⊆ (𝐴 × 𝐵) | |
18 | 16, 17 | eqsstrdi 3969 | . 2 ⊢ (¬ (𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)) |
19 | 14, 18 | pm2.61i 185 | 1 ⊢ ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 × cxp 5517 dom cdm 5519 ran crn 5520 ∘ ccom 5523 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 |
This theorem is referenced by: trclublem 14346 trclubgNEW 40318 trclexi 40320 cnvtrcl0 40326 xpintrreld 40367 trrelsuperreldg 40369 trrelsuperrel2dg 40372 |
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