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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 38765 | . 2 ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) |
3 | trclubNEW.rel | . . . 4 ⊢ (𝜑 → Rel 𝑅) | |
4 | relssdmrn 5901 | . . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅)) |
6 | ssequn1 4012 | . . 3 ⊢ (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅)) | |
7 | 5, 6 | sylib 210 | . 2 ⊢ (𝜑 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅)) |
8 | 2, 7 | sseqtrd 3866 | 1 ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (dom 𝑅 × ran 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 {cab 2811 Vcvv 3414 ∪ cun 3796 ⊆ wss 3798 ∩ cint 4699 × cxp 5344 dom cdm 5346 ran crn 5347 ∘ ccom 5350 Rel wrel 5351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-int 4700 df-br 4876 df-opab 4938 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 |
This theorem is referenced by: (None) |
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