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Mirrors > Home > MPE Home > Th. List > Mathboxes > trclubNEW | Structured version Visualization version GIF version |
Description: If a relation exists then the transitive closure has an upper bound. (Contributed by RP, 24-Jul-2020.) |
Ref | Expression |
---|---|
trclubNEW.rex | ⊢ (𝜑 → 𝑅 ∈ V) |
trclubNEW.rel | ⊢ (𝜑 → Rel 𝑅) |
Ref | Expression |
---|---|
trclubNEW | ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (dom 𝑅 × ran 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trclubNEW.rex | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) | |
2 | 1 | trclubgNEW 43566 | . 2 ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) |
3 | trclubNEW.rel | . . . 4 ⊢ (𝜑 → Rel 𝑅) | |
4 | relssdmrn 6284 | . . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) |
6 | ssequn1 4196 | . . 3 ⊢ (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅)) | |
7 | 5, 6 | sylib 218 | . 2 ⊢ (𝜑 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅)) |
8 | 2, 7 | sseqtrd 4036 | 1 ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (dom 𝑅 × ran 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1535 ∈ wcel 2104 {cab 2710 Vcvv 3477 ∪ cun 3961 ⊆ wss 3963 ∩ cint 4953 × cxp 5681 dom cdm 5683 ran crn 5684 ∘ ccom 5687 Rel wrel 5688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-int 4954 df-br 5150 df-opab 5212 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 |
This theorem is referenced by: (None) |
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