Theorem trclfvcotrg 15065

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 15058	. 2 ⊢ (𝑅 ∈ V → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
2		fvprc 6912	. . 3 ⊢ (¬ 𝑅 ∈ V → (t+‘𝑅) = ∅)
3		0trrel 15030	. . . . 5 ⊢ (∅ ∘ ∅) ⊆ ∅
4	3	a1i 11	. . . 4 ⊢ ((t+‘𝑅) = ∅ → (∅ ∘ ∅) ⊆ ∅)
5		id 22	. . . . 5 ⊢ ((t+‘𝑅) = ∅ → (t+‘𝑅) = ∅)
6	5, 5	coeq12d 5889	. . . 4 ⊢ ((t+‘𝑅) = ∅ → ((t+‘𝑅) ∘ (t+‘𝑅)) = (∅ ∘ ∅))
7	4, 6, 5	3sstr4d 4056	. . 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 182	1 ⊢ ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976  ∅c0 4352   ∘ 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:  cotrcltrcl  43687  brtrclfv2  43689  frege96d  43711  frege97d  43714  frege98d  43715  frege109d  43719  frege131d  43726