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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 Richard Penner, 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 5573 | . . . 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 | sseqtr4d 3605 | . 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑆) |
6 | xptrrel 13567 | . . . 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 5208 | . . 3 ⊢ (𝜑 → (𝑆 ∘ 𝑆) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))) |
9 | 7, 8, 4 | 3sstr4d 3611 | . 2 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆) |
10 | 5, 9 | jca 553 | 1 ⊢ (𝜑 → (𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ⊆ wss 3540 × cxp 5036 dom cdm 5038 ran crn 5039 ∘ ccom 5042 Rel wrel 5043 |
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: (None) |
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