Theorem reltrclfv 13758

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 13748	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
2	1	adantr 481	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3		simpr 477	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → Rel 𝑅)
4		relssdmrn 5656	. . . . 5 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
5		ssequn1 3783	. . . . . 6 ⊢ (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
6	5	biimpi 206	. . . . 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 3641	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (dom 𝑅 × ran 𝑅))
9		xpss 5226	. . 3 ⊢ (dom 𝑅 × ran 𝑅) ⊆ (V × V)
10	8, 9	syl6ss 3615	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (t+‘𝑅) ⊆ (V × V))
11		df-rel 5121	. 2 ⊢ (Rel (t+‘𝑅) ↔ (t+‘𝑅) ⊆ (V × V))
12	10, 11	sylibr 224	1 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → Rel (t+‘𝑅))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572   ⊆ wss 3574   × cxp 5112  dom cdm 5114  ran crn 5115  Rel wrel 5119  ‘cfv 5888  t+ctcl 13724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-trcl 13726

This theorem is referenced by:  frege124d  38053  frege129d  38055  frege133d  38057