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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 5656 | . . . 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 3642 | . 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑆) |
6 | xptrrel 13719 | . . . 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 5286 | . . 3 ⊢ (𝜑 → (𝑆 ∘ 𝑆) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))) |
9 | 7, 8, 4 | 3sstr4d 3648 | . 2 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆) |
10 | 5, 9 | jca 554 | 1 ⊢ (𝜑 → (𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ⊆ wss 3574 × cxp 5112 dom cdm 5114 ran crn 5115 ∘ ccom 5118 Rel wrel 5119 |
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: (None) |
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