Theorem trrelsuperreldg 44471

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 6273	. . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
3	1, 2	syl 18	. . 3 ⊢ (𝜑 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
4		trrelsuperreldg.s	. . 3 ⊢ (𝜑 → 𝑆 = (dom 𝑅 × ran 𝑅))
5	3, 4	sseqtrrd 3975	. 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑆)
6		xptrrel 15045	. . . 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 5852	. . 3 ⊢ (𝜑 → (𝑆 ∘ 𝑆) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
9	7, 8, 4	3sstr4d 3993	. 2 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
10	5, 9	jca 521	1 ⊢ (𝜑 → (𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ⊆ wss 3906   × cxp 5661  dom cdm 5663  ran crn 5664   ∘ ccom 5667  Rel wrel 5668

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-11 2195  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675

This theorem is used by: (None)