Theorem trrelsuperrel2dg 40730

Description: Concrete construction of a superclass of relation 𝑅 which is a transitive relation. (Contributed by RP, 20-Jul-2020.)

Hypothesis

Ref	Expression
trrelsuperrel2dg.s	⊢ (𝜑 → 𝑆 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)))

Assertion

Ref	Expression
trrelsuperrel2dg	⊢ (𝜑 → (𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆))

Proof of Theorem trrelsuperrel2dg

Step	Hyp	Ref	Expression
1		ssun1 4073	. . 3 ⊢ 𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
2		trrelsuperrel2dg.s	. . 3 ⊢ (𝜑 → 𝑆 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3	1, 2	sseqtrrid 3941	. 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑆)
4		xptrrel 14372	. . . . 5 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅)
5		ssun2 4074	. . . . 5 ⊢ (dom 𝑅 × ran 𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
6	4, 5	sstri 3897	. . . 4 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
7	6	a1i 11	. . 3 ⊢ (𝜑 → ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
8	2, 2	coeq12d 5697	. . . 4 ⊢ (𝜑 → (𝑆 ∘ 𝑆) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
9		coundir 6071	. . . . . 6 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = ((𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ∪ ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
10		relcnv 5932	. . . . . . 7 ⊢ Rel ◡◡𝑅
11		cocnvcnv1 6080	. . . . . . . . 9 ⊢ (◡◡𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = (𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
12		relssdmrn 6091	. . . . . . . . . . 11 ⊢ (Rel ◡◡𝑅 → ◡◡𝑅 ⊆ (dom ◡◡𝑅 × ran ◡◡𝑅))
13		dmcnvcnv 5767	. . . . . . . . . . . 12 ⊢ dom ◡◡𝑅 = dom 𝑅
14		rncnvcnv 5768	. . . . . . . . . . . 12 ⊢ ran ◡◡𝑅 = ran 𝑅
15	13, 14	xpeq12i 5545	. . . . . . . . . . 11 ⊢ (dom ◡◡𝑅 × ran ◡◡𝑅) = (dom 𝑅 × ran 𝑅)
16	12, 15	sseqtrdi 3938	. . . . . . . . . 10 ⊢ (Rel ◡◡𝑅 → ◡◡𝑅 ⊆ (dom 𝑅 × ran 𝑅))
17		coss1 5688	. . . . . . . . . 10 ⊢ (◡◡𝑅 ⊆ (dom 𝑅 × ran 𝑅) → (◡◡𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
18	16, 17	syl 17	. . . . . . . . 9 ⊢ (Rel ◡◡𝑅 → (◡◡𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
19	11, 18	eqsstrrid 3937	. . . . . . . 8 ⊢ (Rel ◡◡𝑅 → (𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
20		ssequn1 4081	. . . . . . . 8 ⊢ ((𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ↔ ((𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ∪ ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))) = ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
21	19, 20	sylib 221	. . . . . . 7 ⊢ (Rel ◡◡𝑅 → ((𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ∪ ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))) = ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
22	10, 21	ax-mp 5	. . . . . 6 ⊢ ((𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ∪ ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))) = ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
23	9, 22	eqtri 2782	. . . . 5 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
24		coundi 6070	. . . . . 6 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = (((dom 𝑅 × ran 𝑅) ∘ 𝑅) ∪ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
25		cocnvcnv2 6081	. . . . . . . . 9 ⊢ ((dom 𝑅 × ran 𝑅) ∘ ◡◡𝑅) = ((dom 𝑅 × ran 𝑅) ∘ 𝑅)
26		coss2 5689	. . . . . . . . . 10 ⊢ (◡◡𝑅 ⊆ (dom 𝑅 × ran 𝑅) → ((dom 𝑅 × ran 𝑅) ∘ ◡◡𝑅) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
27	16, 26	syl 17	. . . . . . . . 9 ⊢ (Rel ◡◡𝑅 → ((dom 𝑅 × ran 𝑅) ∘ ◡◡𝑅) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
28	25, 27	eqsstrrid 3937	. . . . . . . 8 ⊢ (Rel ◡◡𝑅 → ((dom 𝑅 × ran 𝑅) ∘ 𝑅) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
29		ssequn1 4081	. . . . . . . 8 ⊢ (((dom 𝑅 × ran 𝑅) ∘ 𝑅) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ↔ (((dom 𝑅 × ran 𝑅) ∘ 𝑅) ∪ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
30	28, 29	sylib 221	. . . . . . 7 ⊢ (Rel ◡◡𝑅 → (((dom 𝑅 × ran 𝑅) ∘ 𝑅) ∪ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
31	10, 30	ax-mp 5	. . . . . 6 ⊢ (((dom 𝑅 × ran 𝑅) ∘ 𝑅) ∪ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
32	24, 31	eqtri 2782	. . . . 5 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
33	23, 32	eqtri 2782	. . . 4 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
34	8, 33	eqtrdi 2810	. . 3 ⊢ (𝜑 → (𝑆 ∘ 𝑆) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
35	7, 34, 2	3sstr4d 3935	. 2 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
36	3, 35	jca 516	1 ⊢ (𝜑 → (𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∪ cun 3852   ⊆ wss 3854   × cxp 5515  ^◡ccnv 5516  dom cdm 5517  ran crn 5518   ∘ ccom 5521  Rel wrel 5522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pr 5291

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-ral 3073  df-rex 3074  df-rab 3077  df-v 3409  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-nul 4222  df-if 4414  df-sn 4516  df-pr 4518  df-op 4522  df-br 5026  df-opab 5088  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529

This theorem is referenced by: (None)