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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 3833 | . . . . . . . 8 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ dom (𝐴 × 𝐵) | |
2 | dmxpss 5565 | . . . . . . . 8 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴 | |
3 | 1, 2 | sstri 3612 | . . . . . . 7 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ 𝐴 |
4 | inss2 3834 | . . . . . . . 8 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵) | |
5 | rnxpss 5566 | . . . . . . . 8 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 | |
6 | 4, 5 | sstri 3612 | . . . . . . 7 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ 𝐵 |
7 | 3, 6 | ssini 3836 | . . . . . 6 ⊢ (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ (𝐴 ∩ 𝐵) |
8 | eqimss 3657 | . . . . . 6 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∩ 𝐵) ⊆ ∅) | |
9 | 7, 8 | syl5ss 3614 | . . . . 5 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (dom (𝐴 × 𝐵) ∩ ran (𝐴 × 𝐵)) ⊆ ∅) |
10 | ss0 3974 | . . . . 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 13718 | . . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = ∅) |
13 | 0ss 3972 | . . 3 ⊢ ∅ ⊆ (𝐴 × 𝐵) | |
14 | 12, 13 | syl6eqss 3655 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)) |
15 | df-ne 2795 | . . . . 5 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ¬ (𝐴 ∩ 𝐵) = ∅) | |
16 | 15 | biimpri 218 | . . . 4 ⊢ (¬ (𝐴 ∩ 𝐵) = ∅ → (𝐴 ∩ 𝐵) ≠ ∅) |
17 | 16 | xpcoidgend 13714 | . . 3 ⊢ (¬ (𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵)) |
18 | ssid 3624 | . . 3 ⊢ (𝐴 × 𝐵) ⊆ (𝐴 × 𝐵) | |
19 | 17, 18 | syl6eqss 3655 | . 2 ⊢ (¬ (𝐴 ∩ 𝐵) = ∅ → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)) |
20 | 14, 19 | pm2.61i 176 | 1 ⊢ ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1483 ≠ wne 2794 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 × cxp 5112 dom cdm 5114 ran crn 5115 ∘ ccom 5118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 |
This theorem is referenced by: trclublem 13734 trclubgNEW 37925 trclexi 37927 cnvtrcl0 37933 xpintrreld 37958 trrelsuperreldg 37960 trrelsuperrel2dg 37963 |
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