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Theorem trclubNEW 43223
Description: If a relation exists then the transitive closure has an upper bound. (Contributed by RP, 24-Jul-2020.)
Hypotheses
Ref Expression
trclubNEW.rex (𝜑𝑅 ∈ V)
trclubNEW.rel (𝜑 → Rel 𝑅)
Assertion
Ref Expression
trclubNEW (𝜑 {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (dom 𝑅 × ran 𝑅))
Distinct variable group:   𝑥,𝑅
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem trclubNEW
StepHypRef Expression
1 trclubNEW.rex . . 3 (𝜑𝑅 ∈ V)
21trclubgNEW 43222 . 2 (𝜑 {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3 trclubNEW.rel . . . 4 (𝜑 → Rel 𝑅)
4 relssdmrn 6278 . . . 4 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
53, 4syl 17 . . 3 (𝜑𝑅 ⊆ (dom 𝑅 × ran 𝑅))
6 ssequn1 4180 . . 3 (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
75, 6sylib 217 . 2 (𝜑 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
82, 7sseqtrd 4019 1 (𝜑 {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (dom 𝑅 × ran 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  {cab 2702  Vcvv 3461  cun 3944  wss 3946   cint 4953   × cxp 5679  dom cdm 5681  ran crn 5682  ccom 5685  Rel wrel 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-br 5153  df-opab 5215  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693
This theorem is referenced by: (None)
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