Theorem reltrclfv 14960

Description: The transitive closure of a relation is a relation. (Contributed by RP, 9-May-2020.)

Assertion

Ref	Expression
reltrclfv	⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → Rel (t+‘𝑅))

Proof of Theorem reltrclfv

Step	Hyp	Ref	Expression
1		trclfvub 14950	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
2	1	adantr 480	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3		simpr 484	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → Rel 𝑅)
4		relssdmrn 6257	. . . . 5 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
5		ssequn1 4172	. . . . . 6 ⊢ (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
6	5	biimpi 215	. . . . 5 ⊢ (𝑅 ⊆ (dom 𝑅 × ran 𝑅) → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
7	3, 4, 6	3syl 18	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
8	2, 7	sseqtrd 4014	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (dom 𝑅 × ran 𝑅))
9		xpss 5682	. . 3 ⊢ (dom 𝑅 × ran 𝑅) ⊆ (V × V)
10	8, 9	sstrdi 3986	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (V × V))
11		df-rel 5673	. 2 ⊢ (Rel (t+‘𝑅) ↔ (t+‘𝑅) ⊆ (V × V))
12	10, 11	sylibr 233	1 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → Rel (t+‘𝑅))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3466   ∪ cun 3938   ⊆ wss 3940   × cxp 5664  dom cdm 5666  ran crn 5667  Rel wrel 5671  ‘cfv 6533  t+ctcl 14928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-iota 6485  df-fun 6535  df-fv 6541  df-trcl 14930

This theorem is referenced by:  frege124d  42967  frege129d  42969  frege133d  42971