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Theorem trclubgNEW 36944
 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)
2 dmexg 6989 . . . . 5 (𝑅 ∈ V → dom 𝑅 ∈ V)
31, 2syl 17 . . . 4 (𝜑 → dom 𝑅 ∈ V)
4 rnexg 6990 . . . . 5 (𝑅 ∈ V → ran 𝑅 ∈ V)
51, 4syl 17 . . . 4 (𝜑 → ran 𝑅 ∈ V)
6 xpexg 6858 . . . 4 ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 × ran 𝑅) ∈ V)
73, 5, 6syl2anc 691 . . 3 (𝜑 → (dom 𝑅 × ran 𝑅) ∈ V)
8 unexg 6857 . . 3 ((𝑅 ∈ V ∧ (dom 𝑅 × ran 𝑅) ∈ V) → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
91, 7, 8syl2anc 691 . 2 (𝜑 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
10 id 22 . . . 4 (𝑥 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)) → 𝑥 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
1110, 10coeq12d 5208 . . 3 (𝑥 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)) → (𝑥𝑥) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
1211, 10sseq12d 3597 . 2 (𝑥 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)) → ((𝑥𝑥) ⊆ 𝑥 ↔ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
13 ssun1 3738 . . 3 𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
1413a1i 11 . 2 (𝜑𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
15 cnvssrndm 5574 . . 3 𝑅 ⊆ (ran 𝑅 × dom 𝑅)
16 coundi 5553 . . . 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)))
17 cnvss 5216 . . . . . . . 8 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → 𝑅(ran 𝑅 × dom 𝑅))
18 coss2 5200 . . . . . . . 8 (𝑅(ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅) ⊆ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (ran 𝑅 × dom 𝑅)))
1917, 18syl 17 . . . . . . 7 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅) ⊆ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (ran 𝑅 × dom 𝑅)))
20 cocnvcnv2 5564 . . . . . . 7 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅)
21 cnvxp 5470 . . . . . . . 8 (ran 𝑅 × dom 𝑅) = (dom 𝑅 × ran 𝑅)
2221coeq2i 5204 . . . . . . 7 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (ran 𝑅 × dom 𝑅)) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅))
2319, 20, 223sstr3g 3608 . . . . . 6 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅) ⊆ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅)))
24 ssequn1 3745 . . . . . 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)))
2523, 24sylib 207 . . . . 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)))
26 coundir 5554 . . . . . 6 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅)) = ((𝑅 ∘ (dom 𝑅 × ran 𝑅)) ∪ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
27 coss1 5199 . . . . . . . . . 10 (𝑅(ran 𝑅 × dom 𝑅) → (𝑅 ∘ (dom 𝑅 × ran 𝑅)) ⊆ ((ran 𝑅 × dom 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
2817, 27syl 17 . . . . . . . . 9 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → (𝑅 ∘ (dom 𝑅 × ran 𝑅)) ⊆ ((ran 𝑅 × dom 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
29 cocnvcnv1 5563 . . . . . . . . 9 (𝑅 ∘ (dom 𝑅 × ran 𝑅)) = (𝑅 ∘ (dom 𝑅 × ran 𝑅))
3021coeq1i 5203 . . . . . . . . 9 ((ran 𝑅 × dom 𝑅) ∘ (dom 𝑅 × ran 𝑅)) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
3128, 29, 303sstr3g 3608 . . . . . . . 8 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → (𝑅 ∘ (dom 𝑅 × ran 𝑅)) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
32 ssequn1 3745 . . . . . . . 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)))
3331, 32sylib 207 . . . . . . 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)))
34 xptrrel 13567 . . . . . . . . 9 ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅)
35 ssun2 3739 . . . . . . . . 9 (dom 𝑅 × ran 𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
3634, 35sstri 3577 . . . . . . . 8 ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
3736a1i 11 . . . . . . 7 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3833, 37eqsstrd 3602 . . . . . 6 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∘ (dom 𝑅 × ran 𝑅)) ∪ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3926, 38syl5eqss 3612 . . . . 5 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
4025, 39eqsstrd 3602 . . . 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)))
4116, 40syl5eqss 3612 . . 3 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
4215, 41mp1i 13 . 2 (𝜑 → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
439, 12, 14, 42clublem 36936 1 (𝜑 {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  ∩ cint 4410   × cxp 5036  ◡ccnv 5037  dom cdm 5038  ran crn 5039   ∘ ccom 5042 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050 This theorem is referenced by:  trclubNEW  36945
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