cnvtrcl0 - Mathbox for Richard Penner

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Theorem cnvtrcl0 43619

Description: The converse of the transitive closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)

**Assertion**
Ref	Expression
cnvtrcl0	⊢ (𝑋 ∈ 𝑉 → ^◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = ∩ {𝑦 ∣ (^◡𝑋 ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦)})

Distinct variable groups: 𝑥,𝑦,𝑉 𝑥,𝑋,𝑦

**Proof of Theorem cnvtrcl0**
Step	Hyp	Ref	Expression
1		cnvco 5828	. . . . . 6 ⊢ ^◡(𝑦 ∘ 𝑦) = (^◡𝑦 ∘ ^◡𝑦)
2		cnvss 5815	. . . . . 6 ⊢ ((𝑦 ∘ 𝑦) ⊆ 𝑦 → ^◡(𝑦 ∘ 𝑦) ⊆ ^◡𝑦)
3	1, 2	eqsstrrid 3975	. . . . 5 ⊢ ((𝑦 ∘ 𝑦) ⊆ 𝑦 → (^◡𝑦 ∘ ^◡𝑦) ⊆ ^◡𝑦)
4		coundir 6197	. . . . . . 7 ⊢ ((^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) = ((^◡𝑦 ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ∪ ((𝑋 ∖ ^◡^◡𝑋) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))))
5		coundi 6196	. . . . . . . . 9 ⊢ (^◡𝑦 ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) = ((^◡𝑦 ∘ ^◡𝑦) ∪ (^◡𝑦 ∘ (𝑋 ∖ ^◡^◡𝑋)))
6		ssid 3958	. . . . . . . . . 10 ⊢ (^◡𝑦 ∘ ^◡𝑦) ⊆ (^◡𝑦 ∘ ^◡𝑦)
7		cononrel2 43588	. . . . . . . . . . 11 ⊢ (^◡𝑦 ∘ (𝑋 ∖ ^◡^◡𝑋)) = ∅
8		0ss 4351	. . . . . . . . . . 11 ⊢ ∅ ⊆ (^◡𝑦 ∘ ^◡𝑦)
9	7, 8	eqsstri 3982	. . . . . . . . . 10 ⊢ (^◡𝑦 ∘ (𝑋 ∖ ^◡^◡𝑋)) ⊆ (^◡𝑦 ∘ ^◡𝑦)
10	6, 9	unssi 4142	. . . . . . . . 9 ⊢ ((^◡𝑦 ∘ ^◡𝑦) ∪ (^◡𝑦 ∘ (𝑋 ∖ ^◡^◡𝑋))) ⊆ (^◡𝑦 ∘ ^◡𝑦)
11	5, 10	eqsstri 3982	. . . . . . . 8 ⊢ (^◡𝑦 ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ⊆ (^◡𝑦 ∘ ^◡𝑦)
12		cononrel1 43587	. . . . . . . . 9 ⊢ ((𝑋 ∖ ^◡^◡𝑋) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) = ∅
13	12, 8	eqsstri 3982	. . . . . . . 8 ⊢ ((𝑋 ∖ ^◡^◡𝑋) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ⊆ (^◡𝑦 ∘ ^◡𝑦)
14	11, 13	unssi 4142	. . . . . . 7 ⊢ ((^◡𝑦 ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ∪ ((𝑋 ∖ ^◡^◡𝑋) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))) ⊆ (^◡𝑦 ∘ ^◡𝑦)
15	4, 14	eqsstri 3982	. . . . . 6 ⊢ ((^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ⊆ (^◡𝑦 ∘ ^◡𝑦)
16		id 22	. . . . . 6 ⊢ ((^◡𝑦 ∘ ^◡𝑦) ⊆ ^◡𝑦 → (^◡𝑦 ∘ ^◡𝑦) ⊆ ^◡𝑦)
17	15, 16	sstrid 3947	. . . . 5 ⊢ ((^◡𝑦 ∘ ^◡𝑦) ⊆ ^◡𝑦 → ((^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ⊆ ^◡𝑦)
18		ssun3 4131	. . . . 5 ⊢ (((^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ⊆ ^◡𝑦 → ((^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ⊆ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
19	3, 17, 18	3syl 18	. . . 4 ⊢ ((𝑦 ∘ 𝑦) ⊆ 𝑦 → ((^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ⊆ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
20		id 22	. . . . . 6 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → 𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
21	20, 20	coeq12d 5807	. . . . 5 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → (𝑥 ∘ 𝑥) = ((^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))))
22	21, 20	sseq12d 3969	. . . 4 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → ((𝑥 ∘ 𝑥) ⊆ 𝑥 ↔ ((^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ⊆ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))))
23	19, 22	imbitrrid 246	. . 3 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → ((𝑦 ∘ 𝑦) ⊆ 𝑦 → (𝑥 ∘ 𝑥) ⊆ 𝑥))
24	23	adantl 481	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) → ((𝑦 ∘ 𝑦) ⊆ 𝑦 → (𝑥 ∘ 𝑥) ⊆ 𝑥))
25		cnvco 5828	. . . . 5 ⊢ ^◡(𝑥 ∘ 𝑥) = (^◡𝑥 ∘ ^◡𝑥)
26		cnvss 5815	. . . . 5 ⊢ ((𝑥 ∘ 𝑥) ⊆ 𝑥 → ^◡(𝑥 ∘ 𝑥) ⊆ ^◡𝑥)
27	25, 26	eqsstrrid 3975	. . . 4 ⊢ ((𝑥 ∘ 𝑥) ⊆ 𝑥 → (^◡𝑥 ∘ ^◡𝑥) ⊆ ^◡𝑥)
28		id 22	. . . . . 6 ⊢ (𝑦 = ^◡𝑥 → 𝑦 = ^◡𝑥)
29	28, 28	coeq12d 5807	. . . . 5 ⊢ (𝑦 = ^◡𝑥 → (𝑦 ∘ 𝑦) = (^◡𝑥 ∘ ^◡𝑥))
30	29, 28	sseq12d 3969	. . . 4 ⊢ (𝑦 = ^◡𝑥 → ((𝑦 ∘ 𝑦) ⊆ 𝑦 ↔ (^◡𝑥 ∘ ^◡𝑥) ⊆ ^◡𝑥))
31	27, 30	imbitrrid 246	. . 3 ⊢ (𝑦 = ^◡𝑥 → ((𝑥 ∘ 𝑥) ⊆ 𝑥 → (𝑦 ∘ 𝑦) ⊆ 𝑦))
32	31	adantl 481	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 = ^◡𝑥) → ((𝑥 ∘ 𝑥) ⊆ 𝑥 → (𝑦 ∘ 𝑦) ⊆ 𝑦))
33		id 22	. . . 4 ⊢ (𝑥 = (𝑋 ∪ (dom 𝑋 × ran 𝑋)) → 𝑥 = (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
34	33, 33	coeq12d 5807	. . 3 ⊢ (𝑥 = (𝑋 ∪ (dom 𝑋 × ran 𝑋)) → (𝑥 ∘ 𝑥) = ((𝑋 ∪ (dom 𝑋 × ran 𝑋)) ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))))
35	34, 33	sseq12d 3969	. 2 ⊢ (𝑥 = (𝑋 ∪ (dom 𝑋 × ran 𝑋)) → ((𝑥 ∘ 𝑥) ⊆ 𝑥 ↔ ((𝑋 ∪ (dom 𝑋 × ran 𝑋)) ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))) ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋))))
36		ssun1 4129	. . 3 ⊢ 𝑋 ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋))
37	36	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
38		trclexlem 14901	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∪ (dom 𝑋 × ran 𝑋)) ∈ V)
39		coundir 6197	. . . . 5 ⊢ ((𝑋 ∪ (dom 𝑋 × ran 𝑋)) ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))) = ((𝑋 ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))) ∪ ((dom 𝑋 × ran 𝑋) ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))))
40		coundi 6196	. . . . . . 7 ⊢ (𝑋 ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))) = ((𝑋 ∘ 𝑋) ∪ (𝑋 ∘ (dom 𝑋 × ran 𝑋)))
41		cossxp 6220	. . . . . . . 8 ⊢ (𝑋 ∘ 𝑋) ⊆ (dom 𝑋 × ran 𝑋)
42		cossxp 6220	. . . . . . . . 9 ⊢ (𝑋 ∘ (dom 𝑋 × ran 𝑋)) ⊆ (dom (dom 𝑋 × ran 𝑋) × ran 𝑋)
43		dmxpss 6120	. . . . . . . . . 10 ⊢ dom (dom 𝑋 × ran 𝑋) ⊆ dom 𝑋
44		xpss1 5638	. . . . . . . . . 10 ⊢ (dom (dom 𝑋 × ran 𝑋) ⊆ dom 𝑋 → (dom (dom 𝑋 × ran 𝑋) × ran 𝑋) ⊆ (dom 𝑋 × ran 𝑋))
45	43, 44	ax-mp 5	. . . . . . . . 9 ⊢ (dom (dom 𝑋 × ran 𝑋) × ran 𝑋) ⊆ (dom 𝑋 × ran 𝑋)
46	42, 45	sstri 3945	. . . . . . . 8 ⊢ (𝑋 ∘ (dom 𝑋 × ran 𝑋)) ⊆ (dom 𝑋 × ran 𝑋)
47	41, 46	unssi 4142	. . . . . . 7 ⊢ ((𝑋 ∘ 𝑋) ∪ (𝑋 ∘ (dom 𝑋 × ran 𝑋))) ⊆ (dom 𝑋 × ran 𝑋)
48	40, 47	eqsstri 3982	. . . . . 6 ⊢ (𝑋 ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))) ⊆ (dom 𝑋 × ran 𝑋)
49		coundi 6196	. . . . . . 7 ⊢ ((dom 𝑋 × ran 𝑋) ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))) = (((dom 𝑋 × ran 𝑋) ∘ 𝑋) ∪ ((dom 𝑋 × ran 𝑋) ∘ (dom 𝑋 × ran 𝑋)))
50		cossxp 6220	. . . . . . . . 9 ⊢ ((dom 𝑋 × ran 𝑋) ∘ 𝑋) ⊆ (dom 𝑋 × ran (dom 𝑋 × ran 𝑋))
51		rnxpss 6121	. . . . . . . . . 10 ⊢ ran (dom 𝑋 × ran 𝑋) ⊆ ran 𝑋
52		xpss2 5639	. . . . . . . . . 10 ⊢ (ran (dom 𝑋 × ran 𝑋) ⊆ ran 𝑋 → (dom 𝑋 × ran (dom 𝑋 × ran 𝑋)) ⊆ (dom 𝑋 × ran 𝑋))
53	51, 52	ax-mp 5	. . . . . . . . 9 ⊢ (dom 𝑋 × ran (dom 𝑋 × ran 𝑋)) ⊆ (dom 𝑋 × ran 𝑋)
54	50, 53	sstri 3945	. . . . . . . 8 ⊢ ((dom 𝑋 × ran 𝑋) ∘ 𝑋) ⊆ (dom 𝑋 × ran 𝑋)
55		xptrrel 14887	. . . . . . . 8 ⊢ ((dom 𝑋 × ran 𝑋) ∘ (dom 𝑋 × ran 𝑋)) ⊆ (dom 𝑋 × ran 𝑋)
56	54, 55	unssi 4142	. . . . . . 7 ⊢ (((dom 𝑋 × ran 𝑋) ∘ 𝑋) ∪ ((dom 𝑋 × ran 𝑋) ∘ (dom 𝑋 × ran 𝑋))) ⊆ (dom 𝑋 × ran 𝑋)
57	49, 56	eqsstri 3982	. . . . . 6 ⊢ ((dom 𝑋 × ran 𝑋) ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))) ⊆ (dom 𝑋 × ran 𝑋)
58	48, 57	unssi 4142	. . . . 5 ⊢ ((𝑋 ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))) ∪ ((dom 𝑋 × ran 𝑋) ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋)))) ⊆ (dom 𝑋 × ran 𝑋)
59	39, 58	eqsstri 3982	. . . 4 ⊢ ((𝑋 ∪ (dom 𝑋 × ran 𝑋)) ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))) ⊆ (dom 𝑋 × ran 𝑋)
60		ssun2 4130	. . . 4 ⊢ (dom 𝑋 × ran 𝑋) ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋))
61	59, 60	sstri 3945	. . 3 ⊢ ((𝑋 ∪ (dom 𝑋 × ran 𝑋)) ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))) ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋))
62	61	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → ((𝑋 ∪ (dom 𝑋 × ran 𝑋)) ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))) ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
63	24, 32, 35, 37, 38, 62	clcnvlem 43616	1 ⊢ (𝑋 ∈ 𝑉 → ^◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = ∩ {𝑦 ∣ (^◡𝑋 ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦)})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {cab 2707 ∖ cdif 3900 ∪ cun 3901 ⊆ wss 3903 ∅c0 4284 ∩ cint 4896 × cxp 5617 ^◡ccnv 5618 dom cdm 5619 ran crn 5620 ∘ ccom 5623

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3395 df-v 3438 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-iota 6438 df-fun 6484 df-fv 6490 df-1st 7924 df-2nd 7925

This theorem is referenced by: (None)

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