Theorem dftrrel2 39035

Description: Alternate definition of the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.)

Assertion

Ref	Expression
dftrrel2	⊢ ( TrRel 𝑅 ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ Rel 𝑅))

Proof of Theorem dftrrel2

Step	Hyp	Ref	Expression
1		df-trrel 39032	. 2 ⊢ ( TrRel 𝑅 ↔ (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
2		dfrel6 38721	. . . . . 6 ⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
3	2	biimpi 217	. . . . 5 ⊢ (Rel 𝑅 → (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
4	3, 3	coeq12d 5813	. . . 4 ⊢ (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) = (𝑅 ∘ 𝑅))
5	4, 3	sseq12d 3955	. . 3 ⊢ (Rel 𝑅 → (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ↔ (𝑅 ∘ 𝑅) ⊆ 𝑅))
6	5	pm5.32ri 580	. 2 ⊢ ((((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅) ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ Rel 𝑅))
7	1, 6	bitri 276	1 ⊢ ( TrRel 𝑅 ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ Rel 𝑅))

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547   ∩ cin 3889   ⊆ wss 3890   × cxp 5623  dom cdm 5625  ran crn 5626   ∘ ccom 5629  Rel wrel 5630   TrRel wtrrel 38572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-trrel 39032

This theorem is referenced by:  dftrrel3  39036  eltrrelsrel  39039  trreleq  39040  dfeqvrel2  39048