Theorem trrelsuperrel2dg 44185

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 4121	. . 3 ⊢ 𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
2		trrelsuperrel2dg.s	. . 3 ⊢ (𝜑 → 𝑆 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3	1, 2	sseqtrrid 3970	. 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑆)
4		xptrrel 14979	. . . . 5 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅)
5		ssun2 4122	. . . . 5 ⊢ (dom 𝑅 × ran 𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
6	4, 5	sstri 3936	. . . 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 5825	. . . 4 ⊢ (𝜑 → (𝑆 ∘ 𝑆) = ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
9		coundir 6220	. . . . . 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 6079	. . . . . . 7 ⊢ Rel ◡◡𝑅
11		cocnvcnv1 6230	. . . . . . . . 9 ⊢ (◡◡𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = (𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
12		relssdmrn 6241	. . . . . . . . . . 11 ⊢ (Rel ◡◡𝑅 → ◡◡𝑅 ⊆ (dom ◡◡𝑅 × ran ◡◡𝑅))
13		dmcnvcnv 5898	. . . . . . . . . . . 12 ⊢ dom ◡◡𝑅 = dom 𝑅
14		rncnvcnv 5899	. . . . . . . . . . . 12 ⊢ ran ◡◡𝑅 = ran 𝑅
15	13, 14	xpeq12i 5664	. . . . . . . . . . 11 ⊢ (dom ◡◡𝑅 × ran ◡◡𝑅) = (dom 𝑅 × ran 𝑅)
16	12, 15	sseqtrdi 3967	. . . . . . . . . 10 ⊢ (Rel ◡◡𝑅 → ◡◡𝑅 ⊆ (dom 𝑅 × ran 𝑅))
17		coss1 5816	. . . . . . . . . 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 3966	. . . . . . . 8 ⊢ (Rel ◡◡𝑅 → (𝑅 ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))))
20		ssequn1 4129	. . . . . . . 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 220	. . . . . . 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 2775	. . . . 5 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
24		coundi 6219	. . . . . 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 6231	. . . . . . . . 9 ⊢ ((dom 𝑅 × ran 𝑅) ∘ ◡◡𝑅) = ((dom 𝑅 × ran 𝑅) ∘ 𝑅)
26		coss2 5817	. . . . . . . . . 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 3966	. . . . . . . 8 ⊢ (Rel ◡◡𝑅 → ((dom 𝑅 × ran 𝑅) ∘ 𝑅) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
29		ssequn1 4129	. . . . . . . 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 220	. . . . . . 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 2775	. . . . 5 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
33	23, 32	eqtri 2775	. . . 4 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
34	8, 33	eqtrdi 2803	. . 3 ⊢ (𝜑 → (𝑆 ∘ 𝑆) = ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)))
35	7, 34, 2	3sstr4d 3982	. 2 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
36	3, 35	jca 518	1 ⊢ (𝜑 → (𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1550   ∪ cun 3893   ⊆ wss 3895   × cxp 5634  ^◡ccnv 5635  dom cdm 5636  ran crn 5637   ∘ ccom 5640  Rel wrel 5641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-11 2181  ax-ext 2724  ax-sep 5236  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-ne 2948  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-br 5091  df-opab 5153  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648

This theorem is referenced by: (None)