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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 3795 | . . . . . . . 8 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ dom (𝐴 × 𝐵) | |
2 | dmxpss 5484 | . . . . . . . 8 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴 | |
3 | 1, 2 | sstri 3577 | . . . . . . 7 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ 𝐴 |
4 | inss2 3796 | . . . . . . . 8 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵) | |
5 | rnxpss 5485 | . . . . . . . 8 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 | |
6 | 4, 5 | sstri 3577 | . . . . . . 7 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ 𝐵 |
7 | 3, 6 | ssini 3798 | . . . . . 6 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ (𝐴 ∩ 𝐵) |
8 | eqimss 3620 | . . . . . 6 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∩ 𝐵) ⊆ ∅) | |
9 | 7, 8 | syl5ss 3579 | . . . . 5 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ∅) |
10 | ss0 3926 | . . . . 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 13566 | . . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = ∅) |
13 | 0ss 3924 | . . 3 ⊢ ∅ ⊆ (𝐴 × 𝐵) | |
14 | 12, 13 | syl6eqss 3618 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)) |
15 | df-ne 2782 | . . . . 5 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ¬ (𝐴 ∩ 𝐵) = ∅) | |
16 | 15 | biimpri 217 | . . . 4 ⊢ (¬ (𝐴 ∩ 𝐵) = ∅ → (𝐴 ∩ 𝐵) ≠ ∅) |
17 | 16 | xpcoidgend 13562 | . . 3 ⊢ (¬ (𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵)) |
18 | ssid 3587 | . . 3 ⊢ (𝐴 × 𝐵) ⊆ (𝐴 × 𝐵) | |
19 | 17, 18 | syl6eqss 3618 | . 2 ⊢ (¬ (𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)) |
20 | 14, 19 | pm2.61i 175 | 1 ⊢ ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ≠ wne 2780 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 × cxp 5036 dom cdm 5038 ran crn 5039 ∘ ccom 5042 |
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-ne 2782 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-dm 5048 df-rn 5049 df-res 5050 |
This theorem is referenced by: trclublem 13582 trclubgNEW 36944 trclexi 36946 cnvtrcl0 36952 xpintrreld 36977 trrelsuperreldg 36979 trrelsuperrel2dg 36982 |
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