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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 40068 | . 2 ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) |
3 | trclubNEW.rel | . . . 4 ⊢ (𝜑 → Rel 𝑅) | |
4 | relssdmrn 6107 | . . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) |
6 | ssequn1 4144 | . . 3 ⊢ (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅)) | |
7 | 5, 6 | sylib 220 | . 2 ⊢ (𝜑 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅)) |
8 | 2, 7 | sseqtrd 3995 | 1 ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (dom 𝑅 × ran 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {cab 2799 Vcvv 3486 ∪ cun 3922 ⊆ wss 3924 ∩ cint 4862 × cxp 5539 dom cdm 5541 ran crn 5542 ∘ ccom 5545 Rel wrel 5546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-int 4863 df-br 5053 df-opab 5115 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 |
This theorem is referenced by: (None) |
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