Theorem trrelsuperreldg 43571

Description: Concrete construction of a superclass of relation 𝑅 which is a transitive relation. (Contributed by RP, 25-Dec-2019.)

Hypotheses

Ref	Expression
trrelsuperreldg.r	⊢ (𝜑 → Rel 𝑅)
trrelsuperreldg.s	⊢ (𝜑 → 𝑆 = (dom 𝑅 × ran 𝑅))

Assertion

Ref	Expression
trrelsuperreldg	⊢ (𝜑 → (𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆))

Proof of Theorem trrelsuperreldg

Step	Hyp	Ref	Expression
1		trrelsuperreldg.r	. . . 4 ⊢ (𝜑 → Rel 𝑅)
2		relssdmrn 6298	. . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
4		trrelsuperreldg.s	. . 3 ⊢ (𝜑 → 𝑆 = (dom 𝑅 × ran 𝑅))
5	3, 4	sseqtrrd 4044	. 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑆)
6		xptrrel 15025	. . . 4 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅)
7	6	a1i 11	. . 3 ⊢ (𝜑 → ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅))
8	4, 4	coeq12d 5888	. . 3 ⊢ (𝜑 → (𝑆 ∘ 𝑆) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
9	7, 8, 4	3sstr4d 4050	. 2 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
10	5, 9	jca 511	1 ⊢ (𝜑 → (𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ⊆ wss 3970   × cxp 5697  dom cdm 5699  ran crn 5700   ∘ ccom 5703  Rel wrel 5704

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 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450

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-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711

This theorem is referenced by: (None)