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Theorem trclubNEW 43567
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 43566 . 2 (𝜑 {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3 trclubNEW.rel . . . 4 (𝜑 → Rel 𝑅)
4 relssdmrn 6284 . . . 4 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
53, 4syl 17 . . 3 (𝜑𝑅 ⊆ (dom 𝑅 × ran 𝑅))
6 ssequn1 4196 . . 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 4036 1 (𝜑 {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 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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