Theorem trclubgNEW 43937

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 7848	. . . 4 ⊢ (𝜑 → dom 𝑅 ∈ V)
3		rnexg 7847	. . . . 5 ⊢ (𝑅 ∈ V → ran 𝑅 ∈ V)
4	1, 3	syl 17	. . . 4 ⊢ (𝜑 → ran 𝑅 ∈ V)
5	2, 4	xpexd 7699	. . 3 ⊢ (𝜑 → (dom 𝑅 × ran 𝑅) ∈ V)
6	1, 5	unexd 7702	. 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 5814	. . 3 ⊢ (𝑥 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)) → (𝑥 ∘ 𝑥) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
9	8, 7	sseq12d 3968	. 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 4131	. . 3 ⊢ 𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
11	10	a1i 11	. 2 ⊢ (𝜑 → 𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
12		cnvssrndm 6230	. . 3 ⊢ ◡𝑅 ⊆ (ran 𝑅 × dom 𝑅)
13		coundi 6206	. . . 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 5822	. . . . . . . 8 ⊢ (◡𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ◡◡𝑅 ⊆ ◡(ran 𝑅 × dom 𝑅))
15		coss2 5806	. . . . . . . 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 6218	. . . . . . 7 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ ◡◡𝑅) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅)
18		cnvxp 6116	. . . . . . . 8 ⊢ ◡(ran 𝑅 × dom 𝑅) = (dom 𝑅 × ran 𝑅)
19	18	coeq2i 5810	. . . . . . 7 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ ◡(ran 𝑅 × dom 𝑅)) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅))
20	16, 17, 19	3sstr3g 3987	. . . . . 6 ⊢ (◡𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ 𝑅) ⊆ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅)))
21		ssequn1 4139	. . . . . 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 6207	. . . . . 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 5805	. . . . . . . . . 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 6217	. . . . . . . . 9 ⊢ (◡◡𝑅 ∘ (dom 𝑅 × ran 𝑅)) = (𝑅 ∘ (dom 𝑅 × ran 𝑅))
27	18	coeq1i 5809	. . . . . . . . 9 ⊢ (◡(ran 𝑅 × dom 𝑅) ∘ (dom 𝑅 × ran 𝑅)) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
28	25, 26, 27	3sstr3g 3987	. . . . . . . 8 ⊢ (◡𝑅 ⊆ (ran 𝑅 × dom 𝑅) → (𝑅 ∘ (dom 𝑅 × ran 𝑅)) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
29		ssequn1 4139	. . . . . . . 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 14908	. . . . . . . . 9 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅)
32		ssun2 4132	. . . . . . . . 9 ⊢ (dom 𝑅 × ran 𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
33	31, 32	sstri 3944	. . . . . . . 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 3969	. . . . . 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 3973	. . . . 5 ⊢ (◡𝑅 ⊆ (ran 𝑅 × dom 𝑅) → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
37	22, 36	eqsstrd 3969	. . . 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 3973	. . 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 43929	1 ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  Vcvv 3441   ∪ cun 3900   ⊆ wss 3902  ∩ cint 4903   × cxp 5623  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   ∘ ccom 5629

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-br 5100  df-opab 5162  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637

This theorem is referenced by:  trclubNEW  43938