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	⊢ (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		dmexg 6989	. . . . 5 ⊢ (𝑅 ∈ V → dom 𝑅 ∈ V)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → dom 𝑅 ∈ V)
4		rnexg 6990	. . . . 5 ⊢ (𝑅 ∈ V → ran 𝑅 ∈ V)
5	1, 4	syl 17	. . . 4 ⊢ (𝜑 → ran 𝑅 ∈ V)
6		xpexg 6858	. . . 4 ⊢ ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 × ran 𝑅) ∈ V)
7	3, 5, 6	syl2anc 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)
9	1, 7, 8	syl2anc 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)))
11	10, 10	coeq12d 5208	. . 3 ⊢ (𝑥 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)) → (𝑥 ∘ 𝑥) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
12	11, 10	sseq12d 3597	. 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))))
13		ssun1 3738	. . 3 ⊢ 𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
14	13	a1i 11	. 2 ⊢ (𝜑 → 𝑅 ⊆ (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)))
19	17, 18	syl 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)
22	21	coeq2i 5204	. . . . . . 7 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ ◡(ran 𝑅 × dom 𝑅)) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅))
23	19, 20, 22	3sstr3g 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)))
25	23, 24	sylib 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)))
28	17, 27	syl 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))
30	21	coeq1i 5203	. . . . . . . . 9 ⊢ (◡(ran 𝑅 × dom 𝑅) ∘ (dom 𝑅 × ran 𝑅)) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
31	28, 29, 30	3sstr3g 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)))
33	31, 32	sylib 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))
36	34, 35	sstri 3577	. . . . . . . 8 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
37	36	a1i 11	. . . . . . 7 ⊢ (◡𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
38	33, 37	eqsstrd 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)))
39	26, 38	syl5eqss 3612	. . . . 5 ⊢ (◡𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
40	25, 39	eqsstrd 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)))
41	16, 40	syl5eqss 3612	. . 3 ⊢ (◡𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
42	15, 41	mp1i 13	. 2 ⊢ (𝜑 → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
43	9, 12, 14, 42	clublem 36936	1 ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))

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