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Theorem trclubgNEW 39839
Description: If a relation exists then the transitive closure has an upper bound. (Contributed by RP, 24-Jul-2020.)
Hypothesis
Ref Expression
trclubgNEW.rex (𝜑𝑅 ∈ V)
Assertion
Ref Expression
trclubgNEW (𝜑 {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
Distinct variable group:   𝑥,𝑅
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem trclubgNEW
StepHypRef Expression
1 trclubgNEW.rex . . 3 (𝜑𝑅 ∈ V)
21dmexd 7606 . . . 4 (𝜑 → dom 𝑅 ∈ V)
3 rnexg 7605 . . . . 5 (𝑅 ∈ V → ran 𝑅 ∈ V)
41, 3syl 17 . . . 4 (𝜑 → ran 𝑅 ∈ V)
52, 4xpexd 7466 . . 3 (𝜑 → (dom 𝑅 × ran 𝑅) ∈ V)
6 unexg 7464 . . 3 ((𝑅 ∈ V ∧ (dom 𝑅 × ran 𝑅) ∈ V) → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
71, 5, 6syl2anc 584 . 2 (𝜑 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
8 id 22 . . . 4 (𝑥 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)) → 𝑥 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
98, 8coeq12d 5733 . . 3 (𝑥 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)) → (𝑥𝑥) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
109, 8sseq12d 4003 . 2 (𝑥 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)) → ((𝑥𝑥) ⊆ 𝑥 ↔ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
11 ssun1 4151 . . 3 𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
1211a1i 11 . 2 (𝜑𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
13 cnvssrndm 6119 . . 3 𝑅 ⊆ (ran 𝑅 × dom 𝑅)
14 coundi 6097 . . . 4 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = (((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅) ∪ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅)))
15 cnvss 5741 . . . . . . . 8 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → 𝑅(ran 𝑅 × dom 𝑅))
16 coss2 5725 . . . . . . . 8 (𝑅(ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅) ⊆ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (ran 𝑅 × dom 𝑅)))
1715, 16syl 17 . . . . . . 7 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅) ⊆ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (ran 𝑅 × dom 𝑅)))
18 cocnvcnv2 6108 . . . . . . 7 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅)
19 cnvxp 6011 . . . . . . . 8 (ran 𝑅 × dom 𝑅) = (dom 𝑅 × ran 𝑅)
2019coeq2i 5729 . . . . . . 7 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (ran 𝑅 × dom 𝑅)) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅))
2117, 18, 203sstr3g 4014 . . . . . 6 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅) ⊆ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅)))
22 ssequn1 4159 . . . . . 6 (((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅) ⊆ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅)) ↔ (((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅) ∪ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅))) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅)))
2321, 22sylib 219 . . . . 5 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → (((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅) ∪ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅))) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅)))
24 coundir 6098 . . . . . 6 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅)) = ((𝑅 ∘ (dom 𝑅 × ran 𝑅)) ∪ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
25 coss1 5724 . . . . . . . . . 10 (𝑅(ran 𝑅 × dom 𝑅) → (𝑅 ∘ (dom 𝑅 × ran 𝑅)) ⊆ ((ran 𝑅 × dom 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
2615, 25syl 17 . . . . . . . . 9 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → (𝑅 ∘ (dom 𝑅 × ran 𝑅)) ⊆ ((ran 𝑅 × dom 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
27 cocnvcnv1 6107 . . . . . . . . 9 (𝑅 ∘ (dom 𝑅 × ran 𝑅)) = (𝑅 ∘ (dom 𝑅 × ran 𝑅))
2819coeq1i 5728 . . . . . . . . 9 ((ran 𝑅 × dom 𝑅) ∘ (dom 𝑅 × ran 𝑅)) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
2926, 27, 283sstr3g 4014 . . . . . . . 8 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → (𝑅 ∘ (dom 𝑅 × ran 𝑅)) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
30 ssequn1 4159 . . . . . . . 8 ((𝑅 ∘ (dom 𝑅 × ran 𝑅)) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ↔ ((𝑅 ∘ (dom 𝑅 × ran 𝑅)) ∪ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
3129, 30sylib 219 . . . . . . 7 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∘ (dom 𝑅 × ran 𝑅)) ∪ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
32 xptrrel 14333 . . . . . . . . 9 ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅)
33 ssun2 4152 . . . . . . . . 9 (dom 𝑅 × ran 𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
3432, 33sstri 3979 . . . . . . . 8 ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
3534a1i 11 . . . . . . 7 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3631, 35eqsstrd 4008 . . . . . 6 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∘ (dom 𝑅 × ran 𝑅)) ∪ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3724, 36eqsstrid 4018 . . . . 5 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3823, 37eqsstrd 4008 . . . 4 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → (((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅) ∪ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3914, 38eqsstrid 4018 . . 3 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
4013, 39mp1i 13 . 2 (𝜑 → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
417, 10, 12, 40clublem 39831 1 (𝜑 {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  {cab 2803  Vcvv 3499  cun 3937  wss 3939   cint 4873   × cxp 5551  ccnv 5552  dom cdm 5553  ran crn 5554  ccom 5557
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-int 4874  df-br 5063  df-opab 5125  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565
This theorem is referenced by:  trclubNEW  39840
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