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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 6114 | . . . 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 4005 | . 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑆) |
6 | xptrrel 14328 | . . . 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 5728 | . . 3 ⊢ (𝜑 → (𝑆 ∘ 𝑆) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))) |
9 | 7, 8, 4 | 3sstr4d 4011 | . 2 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆) |
10 | 5, 9 | jca 512 | 1 ⊢ (𝜑 → (𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ⊆ wss 3933 × cxp 5546 dom cdm 5548 ran crn 5549 ∘ ccom 5552 Rel wrel 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 |
This theorem is referenced by: (None) |
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