Theorem trrelsuperreldg 44244

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 6256	. . . 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 3973	. 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑆)
6		xptrrel 14993	. . . 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 5836	. . 3 ⊢ (𝜑 → (𝑆 ∘ 𝑆) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
9	7, 8, 4	3sstr4d 3991	. 2 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
10	5, 9	jca 519	1 ⊢ (𝜑 → (𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ⊆ wss 3904   × cxp 5645  dom cdm 5647  ran crn 5648   ∘ ccom 5651  Rel wrel 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-11 2191  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659

This theorem is referenced by: (None)