Theorem trclubgNEW 43631

Description: If a relation exists then the transitive closure has an upper bound. (Contributed by RP, 24-Jul-2020.)

Hypothesis

Ref	Expression
trclubgNEW.rex	⊢ (𝜑 → 𝑅 ∈ V)

Assertion

Ref	Expression
trclubgNEW	⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))

Distinct variable group:   𝑥,𝑅

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem trclubgNEW

Step	Hyp	Ref	Expression
1		trclubgNEW.rex	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
2	1	dmexd 7925	. . . 4 ⊢ (𝜑 → dom 𝑅 ∈ V)
3		rnexg 7924	. . . . 5 ⊢ (𝑅 ∈ V → ran 𝑅 ∈ V)
4	1, 3	syl 17	. . . 4 ⊢ (𝜑 → ran 𝑅 ∈ V)
5	2, 4	xpexd 7771	. . 3 ⊢ (𝜑 → (dom 𝑅 × ran 𝑅) ∈ V)
6	1, 5	unexd 7774	. 2 ⊢ (𝜑 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
7		id 22	. . . 4 ⊢ (𝑥 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)) → 𝑥 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
8	7, 7	coeq12d 5875	. . 3 ⊢ (𝑥 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)) → (𝑥 ∘ 𝑥) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
9	8, 7	sseq12d 4017	. 2 ⊢ (𝑥 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)) → ((𝑥 ∘ 𝑥) ⊆ 𝑥 ↔ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
10		ssun1 4178	. . 3 ⊢ 𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
11	10	a1i 11	. 2 ⊢ (𝜑 → 𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
12		cnvssrndm 6291	. . 3 ⊢ ◡𝑅 ⊆ (ran 𝑅 × dom 𝑅)
13		coundi 6267	. . . 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)))
14		cnvss 5883	. . . . . . . 8 ⊢ (◡𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ◡◡𝑅 ⊆ ◡(ran 𝑅 × dom 𝑅))
15		coss2 5867	. . . . . . . 8 ⊢ (◡◡𝑅 ⊆ ◡(ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ ◡◡𝑅) ⊆ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ ◡(ran 𝑅 × dom 𝑅)))
16	14, 15	syl 17	. . . . . . 7 ⊢ (◡𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ ◡◡𝑅) ⊆ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ ◡(ran 𝑅 × dom 𝑅)))
17		cocnvcnv2 6278	. . . . . . 7 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ ◡◡𝑅) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅)
18		cnvxp 6177	. . . . . . . 8 ⊢ ◡(ran 𝑅 × dom 𝑅) = (dom 𝑅 × ran 𝑅)
19	18	coeq2i 5871	. . . . . . 7 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ ◡(ran 𝑅 × dom 𝑅)) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅))
20	16, 17, 19	3sstr3g 4036	. . . . . 6 ⊢ (◡𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅) ⊆ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅)))
21		ssequn1 4186	. . . . . 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)))
22	20, 21	sylib 218	. . . . 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)))
23		coundir 6268	. . . . . 6 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅)) = ((𝑅 ∘ (dom 𝑅 × ran 𝑅)) ∪ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
24		coss1 5866	. . . . . . . . . 10 ⊢ (◡◡𝑅 ⊆ ◡(ran 𝑅 × dom 𝑅) → (◡◡𝑅 ∘ (dom 𝑅 × ran 𝑅)) ⊆ (◡(ran 𝑅 × dom 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
25	14, 24	syl 17	. . . . . . . . 9 ⊢ (◡𝑅 ⊆ (ran 𝑅 × dom 𝑅) → (◡◡𝑅 ∘ (dom 𝑅 × ran 𝑅)) ⊆ (◡(ran 𝑅 × dom 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
26		cocnvcnv1 6277	. . . . . . . . 9 ⊢ (◡◡𝑅 ∘ (dom 𝑅 × ran 𝑅)) = (𝑅 ∘ (dom 𝑅 × ran 𝑅))
27	18	coeq1i 5870	. . . . . . . . 9 ⊢ (◡(ran 𝑅 × dom 𝑅) ∘ (dom 𝑅 × ran 𝑅)) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
28	25, 26, 27	3sstr3g 4036	. . . . . . . 8 ⊢ (◡𝑅 ⊆ (ran 𝑅 × dom 𝑅) → (𝑅 ∘ (dom 𝑅 × ran 𝑅)) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
29		ssequn1 4186	. . . . . . . 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)))
30	28, 29	sylib 218	. . . . . . 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)))
31		xptrrel 15019	. . . . . . . . 9 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅)
32		ssun2 4179	. . . . . . . . 9 ⊢ (dom 𝑅 × ran 𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
33	31, 32	sstri 3993	. . . . . . . 8 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
34	33	a1i 11	. . . . . . 7 ⊢ (◡𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
35	30, 34	eqsstrd 4018	. . . . . 6 ⊢ (◡𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∘ (dom 𝑅 × ran 𝑅)) ∪ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
36	23, 35	eqsstrid 4022	. . . . 5 ⊢ (◡𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
37	22, 36	eqsstrd 4018	. . . 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)))
38	13, 37	eqsstrid 4022	. . 3 ⊢ (◡𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
39	12, 38	mp1i 13	. 2 ⊢ (𝜑 → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
40	6, 9, 11, 39	clublem 43623	1 ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  Vcvv 3480   ∪ cun 3949   ⊆ wss 3951  ∩ cint 4946   × cxp 5683  ^◡ccnv 5684  dom cdm 5685  ran crn 5686   ∘ ccom 5689

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 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

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 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697

This theorem is referenced by:  trclubNEW  43632