Theorem cotrtrclfv 15061

Description: The transitive closure of a transitive relation. (Contributed by RP, 28-Apr-2020.)

Assertion

Ref	Expression
cotrtrclfv	⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) → (t+‘𝑅) = 𝑅)

Proof of Theorem cotrtrclfv

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		trclfv 15049	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})
2	1	adantr 480	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) → (t+‘𝑅) = ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})
3		simpr 484	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) → (𝑅 ∘ 𝑅) ⊆ 𝑅)
4		ssid 4031	. . . . . 6 ⊢ 𝑅 ⊆ 𝑅
5	3, 4	jctil 519	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) → (𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅))
6		trcleq2lem 15040	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ↔ (𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅)))
7	6	elabg 3690	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ↔ (𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅)))
8	7	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) → (𝑅 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ↔ (𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅)))
9	5, 8	mpbird 257	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) → 𝑅 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)})
10		intss1 4987	. . . 4 ⊢ (𝑅 ∈ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ⊆ 𝑅)
11	9, 10	syl 17	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ⊆ 𝑅)
12	2, 11	eqsstrd 4047	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) → (t+‘𝑅) ⊆ 𝑅)
13		trclfvlb 15057	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (t+‘𝑅))
14	13	adantr 480	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) → 𝑅 ⊆ (t+‘𝑅))
15	12, 14	eqssd 4026	1 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) → (t+‘𝑅) = 𝑅)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717   ⊆ wss 3976  ∩ cint 4970   ∘ ccom 5704  ‘cfv 6573  t+ctcl 15034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-iota 6525  df-fun 6575  df-fv 6581  df-trcl 15036

This theorem is referenced by:  trclidm  15062