Theorem trclubgNEW 37925

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 7097	. . . . 5 ⊢ (𝑅 ∈ V → dom 𝑅 ∈ V)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → dom 𝑅 ∈ V)
4		rnexg 7098	. . . . 5 ⊢ (𝑅 ∈ V → ran 𝑅 ∈ V)
5	1, 4	syl 17	. . . 4 ⊢ (𝜑 → ran 𝑅 ∈ V)
6		xpexg 6960	. . . 4 ⊢ ((dom 𝑅 ∈ V ∧ ran 𝑅 ∈ V) → (dom 𝑅 × ran 𝑅) ∈ V)
7	3, 5, 6	syl2anc 693	. . 3 ⊢ (𝜑 → (dom 𝑅 × ran 𝑅) ∈ V)
8		unexg 6959	. . 3 ⊢ ((𝑅 ∈ V ∧ (dom 𝑅 × ran 𝑅) ∈ V) → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
9	1, 7, 8	syl2anc 693	. 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 5286	. . 3 ⊢ (𝑥 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)) → (𝑥 ∘ 𝑥) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
12	11, 10	sseq12d 3634	. 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 3776	. . 3 ⊢ 𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
14	13	a1i 11	. 2 ⊢ (𝜑 → 𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
15		cnvssrndm 5657	. . 3 ⊢ ◡𝑅 ⊆ (ran 𝑅 × dom 𝑅)
16		coundi 5636	. . . 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 5294	. . . . . . . 8 ⊢ (◡𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ◡◡𝑅 ⊆ ◡(ran 𝑅 × dom 𝑅))
18		coss2 5278	. . . . . . . 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 5647	. . . . . . 7 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ ◡◡𝑅) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅)
21		cnvxp 5551	. . . . . . . 8 ⊢ ◡(ran 𝑅 × dom 𝑅) = (dom 𝑅 × ran 𝑅)
22	21	coeq2i 5282	. . . . . . 7 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ ◡(ran 𝑅 × dom 𝑅)) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅))
23	19, 20, 22	3sstr3g 3645	. . . . . 6 ⊢ (◡𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅) ⊆ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅)))
24		ssequn1 3783	. . . . . 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 208	. . . . 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 5637	. . . . . 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 5277	. . . . . . . . . 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 5646	. . . . . . . . 9 ⊢ (◡◡𝑅 ∘ (dom 𝑅 × ran 𝑅)) = (𝑅 ∘ (dom 𝑅 × ran 𝑅))
30	21	coeq1i 5281	. . . . . . . . 9 ⊢ (◡(ran 𝑅 × dom 𝑅) ∘ (dom 𝑅 × ran 𝑅)) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
31	28, 29, 30	3sstr3g 3645	. . . . . . . 8 ⊢ (◡𝑅 ⊆ (ran 𝑅 × dom 𝑅) → (𝑅 ∘ (dom 𝑅 × ran 𝑅)) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
32		ssequn1 3783	. . . . . . . 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 208	. . . . . . 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 13719	. . . . . . . . 9 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅)
35		ssun2 3777	. . . . . . . . 9 ⊢ (dom 𝑅 × ran 𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
36	34, 35	sstri 3612	. . . . . . . 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 3639	. . . . . 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 3649	. . . . 5 ⊢ (◡𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
40	25, 39	eqsstrd 3639	. . . 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 3649	. . 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 37917	1 ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  Vcvv 3200   ∪ cun 3572   ⊆ wss 3574  ∩ cint 4475   × cxp 5112  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   ∘ ccom 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by:  trclubNEW  37926