Theorem cotrtrclfv 14935

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 14923	. . . 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 3956	. . . . . 6 ⊢ 𝑅 ⊆ 𝑅
5	3, 4	jctil 519	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) → (𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅))
6		trcleq2lem 14914	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) ↔ (𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅)))
7	6	elabg 3631	. . . . . 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 4918	. . . 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 3968	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) → (t+‘𝑅) ⊆ 𝑅)
13		trclfvlb 14931	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (t+‘𝑅))
14	13	adantr 480	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) → 𝑅 ⊆ (t+‘𝑅))
15	12, 14	eqssd 3951	1 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) → (t+‘𝑅) = 𝑅)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2714   ⊆ wss 3901  ∩ cint 4902   ∘ ccom 5628  ‘cfv 6492  t+ctcl 14908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-iota 6448  df-fun 6494  df-fv 6500  df-trcl 14910

This theorem is referenced by:  trclidm  14936