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Theorem trclubNEW 43610
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 43609 . 2 (𝜑 {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3 trclubNEW.rel . . . 4 (𝜑 → Rel 𝑅)
4 relssdmrn 6286 . . . 4 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
53, 4syl 17 . . 3 (𝜑𝑅 ⊆ (dom 𝑅 × ran 𝑅))
6 ssequn1 4185 . . 3 (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
75, 6sylib 218 . 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 395   = wceq 1540  wcel 2108  {cab 2713  Vcvv 3479  cun 3948  wss 3950   cint 4944   × 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 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-int 4945  df-br 5142  df-opab 5204  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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