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Theorem trclubgNEW 40719
 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 7620 . . . 4 (𝜑 → dom 𝑅 ∈ V)
3 rnexg 7619 . . . . 5 (𝑅 ∈ V → ran 𝑅 ∈ V)
41, 3syl 17 . . . 4 (𝜑 → ran 𝑅 ∈ V)
52, 4xpexd 7477 . . 3 (𝜑 → (dom 𝑅 × ran 𝑅) ∈ V)
6 unexg 7475 . . 3 ((𝑅 ∈ V ∧ (dom 𝑅 × ran 𝑅) ∈ V) → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
71, 5, 6syl2anc 587 . 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 5709 . . 3 (𝑥 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)) → (𝑥𝑥) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
109, 8sseq12d 3927 . 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 4079 . . 3 𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
1211a1i 11 . 2 (𝜑𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
13 cnvssrndm 6104 . . 3 𝑅 ⊆ (ran 𝑅 × dom 𝑅)
14 coundi 6081 . . . 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 5717 . . . . . . . 8 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → 𝑅(ran 𝑅 × dom 𝑅))
16 coss2 5701 . . . . . . . 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 6092 . . . . . . 7 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅)
19 cnvxp 5990 . . . . . . . 8 (ran 𝑅 × dom 𝑅) = (dom 𝑅 × ran 𝑅)
2019coeq2i 5705 . . . . . . 7 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (ran 𝑅 × dom 𝑅)) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅))
2117, 18, 203sstr3g 3938 . . . . . 6 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅) ⊆ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅)))
22 ssequn1 4087 . . . . . 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 221 . . . . 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 6082 . . . . . 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 5700 . . . . . . . . . 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 6091 . . . . . . . . 9 (𝑅 ∘ (dom 𝑅 × ran 𝑅)) = (𝑅 ∘ (dom 𝑅 × ran 𝑅))
2819coeq1i 5704 . . . . . . . . 9 ((ran 𝑅 × dom 𝑅) ∘ (dom 𝑅 × ran 𝑅)) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
2926, 27, 283sstr3g 3938 . . . . . . . 8 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → (𝑅 ∘ (dom 𝑅 × ran 𝑅)) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
30 ssequn1 4087 . . . . . . . 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 221 . . . . . . 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 14392 . . . . . . . . 9 ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅)
33 ssun2 4080 . . . . . . . . 9 (dom 𝑅 × ran 𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
3432, 33sstri 3903 . . . . . . . 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 3932 . . . . . 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 3942 . . . . 5 (𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3823, 37eqsstrd 3932 . . . 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 3942 . . 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 40711 1 (𝜑 {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  Vcvv 3409   ∪ cun 3858   ⊆ wss 3860  ∩ cint 4841   × cxp 5525  ◡ccnv 5526  dom cdm 5527  ran crn 5528   ∘ ccom 5531 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-int 4842  df-br 5036  df-opab 5098  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539 This theorem is referenced by:  trclubNEW  40720
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