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Mirrors > Home > MPE Home > Th. List > Mathboxes > trrelsuperreldg | Structured version Visualization version GIF version |
Description: Concrete construction of a superclass of relation 𝑅 which is a transitive relation. (Contributed by RP, 25-Dec-2019.) |
Ref | Expression |
---|---|
trrelsuperreldg.r | ⊢ (𝜑 → Rel 𝑅) |
trrelsuperreldg.s | ⊢ (𝜑 → 𝑆 = (dom 𝑅 × ran 𝑅)) |
Ref | Expression |
---|---|
trrelsuperreldg | ⊢ (𝜑 → (𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trrelsuperreldg.r | . . . 4 ⊢ (𝜑 → Rel 𝑅) | |
2 | relssdmrn 6182 | . . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) |
4 | trrelsuperreldg.s | . . 3 ⊢ (𝜑 → 𝑆 = (dom 𝑅 × ran 𝑅)) | |
5 | 3, 4 | sseqtrrd 3967 | . 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑆) |
6 | xptrrel 14732 | . . . 4 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅) | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅)) |
8 | 4, 4 | coeq12d 5782 | . . 3 ⊢ (𝜑 → (𝑆 ∘ 𝑆) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))) |
9 | 7, 8, 4 | 3sstr4d 3973 | . 2 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆) |
10 | 5, 9 | jca 513 | 1 ⊢ (𝜑 → (𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1539 ⊆ wss 3892 × cxp 5594 dom cdm 5596 ran crn 5597 ∘ ccom 5600 Rel wrel 5601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pr 5361 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3287 df-v 3439 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4566 df-pr 4568 df-op 4572 df-br 5082 df-opab 5144 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 |
This theorem is referenced by: (None) |
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