Theorem trrelsuperrel2dg 42725

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 4172	. . 3 ⊢ 𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
2		trrelsuperrel2dg.s	. . 3 ⊢ (𝜑 → 𝑆 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3	1, 2	sseqtrrid 4035	. 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑆)
4		xptrrel 14932	. . . . 5 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅)
5		ssun2 4173	. . . . 5 ⊢ (dom 𝑅 × ran 𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
6	4, 5	sstri 3991	. . . 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 5864	. . . 4 ⊢ (𝜑 → (𝑆 ∘ 𝑆) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
9		coundir 6247	. . . . . 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 6103	. . . . . . 7 ⊢ Rel ◡◡𝑅
11		cocnvcnv1 6256	. . . . . . . . 9 ⊢ (◡◡𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = (𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
12		relssdmrn 6267	. . . . . . . . . . 11 ⊢ (Rel ◡◡𝑅 → ◡◡𝑅 ⊆ (dom ◡◡𝑅 × ran ◡◡𝑅))
13		dmcnvcnv 5932	. . . . . . . . . . . 12 ⊢ dom ◡◡𝑅 = dom 𝑅
14		rncnvcnv 5933	. . . . . . . . . . . 12 ⊢ ran ◡◡𝑅 = ran 𝑅
15	13, 14	xpeq12i 5704	. . . . . . . . . . 11 ⊢ (dom ◡◡𝑅 × ran ◡◡𝑅) = (dom 𝑅 × ran 𝑅)
16	12, 15	sseqtrdi 4032	. . . . . . . . . 10 ⊢ (Rel ◡◡𝑅 → ◡◡𝑅 ⊆ (dom 𝑅 × ran 𝑅))
17		coss1 5855	. . . . . . . . . 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 4031	. . . . . . . 8 ⊢ (Rel ◡◡𝑅 → (𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
20		ssequn1 4180	. . . . . . . 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 217	. . . . . . 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 2759	. . . . 5 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
24		coundi 6246	. . . . . 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 6257	. . . . . . . . 9 ⊢ ((dom 𝑅 × ran 𝑅) ∘ ◡◡𝑅) = ((dom 𝑅 × ran 𝑅) ∘ 𝑅)
26		coss2 5856	. . . . . . . . . 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 4031	. . . . . . . 8 ⊢ (Rel ◡◡𝑅 → ((dom 𝑅 × ran 𝑅) ∘ 𝑅) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
29		ssequn1 4180	. . . . . . . 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 217	. . . . . . 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 2759	. . . . 5 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
33	23, 32	eqtri 2759	. . . 4 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
34	8, 33	eqtrdi 2787	. . 3 ⊢ (𝜑 → (𝑆 ∘ 𝑆) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
35	7, 34, 2	3sstr4d 4029	. 2 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
36	3, 35	jca 511	1 ⊢ (𝜑 → (𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∪ cun 3946   ⊆ wss 3948   × cxp 5674  ^◡ccnv 5675  dom cdm 5676  ran crn 5677   ∘ ccom 5680  Rel wrel 5681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688

This theorem is referenced by: (None)