Theorem trclfvcotrg 14579

Description: The value of the transitive closure of a relation is always a transitive relation. (Contributed by RP, 8-May-2020.)

Assertion

Ref	Expression
trclfvcotrg	⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)

Proof of Theorem trclfvcotrg

Step	Hyp	Ref	Expression
1		trclfvcotr 14572	. 2 ⊢ (𝑅 ∈ V → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
2		fvprc 6709	. . 3 ⊢ (¬ 𝑅 ∈ V → (t+‘𝑅) = ∅)
3		0trrel 14544	. . . . 5 ⊢ (∅ ∘ ∅) ⊆ ∅
4	3	a1i 11	. . . 4 ⊢ ((t+‘𝑅) = ∅ → (∅ ∘ ∅) ⊆ ∅)
5		id 22	. . . . 5 ⊢ ((t+‘𝑅) = ∅ → (t+‘𝑅) = ∅)
6	5, 5	coeq12d 5733	. . . 4 ⊢ ((t+‘𝑅) = ∅ → ((t+‘𝑅) ∘ (t+‘𝑅)) = (∅ ∘ ∅))
7	4, 6, 5	3sstr4d 3948	. . 3 ⊢ ((t+‘𝑅) = ∅ → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
8	2, 7	syl 17	. 2 ⊢ (¬ 𝑅 ∈ V → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
9	1, 8	pm2.61i 185	1 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)

Syntax hints:  ¬ wn 3   = wceq 1543   ∈ wcel 2110  Vcvv 3408   ⊆ wss 3866  ∅c0 4237   ∘ ccom 5555  ‘cfv 6380  t+ctcl 14548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-iota 6338  df-fun 6382  df-fv 6388  df-trcl 14550

This theorem is referenced by:  cotrcltrcl  41010  brtrclfv2  41012  frege96d  41034  frege97d  41037  frege98d  41038  frege109d  41042  frege131d  41049