cnvtrcl0 - Mathbox for Richard Penner

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Theorem cnvtrcl0 37933

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 5308	. . . . . 6 ⊢ ^◡(𝑦 ∘ 𝑦) = (^◡𝑦 ∘ ^◡𝑦)
2		cnvss 5294	. . . . . 6 ⊢ ((𝑦 ∘ 𝑦) ⊆ 𝑦 → ^◡(𝑦 ∘ 𝑦) ⊆ ^◡𝑦)
3	1, 2	syl5eqssr 3650	. . . . 5 ⊢ ((𝑦 ∘ 𝑦) ⊆ 𝑦 → (^◡𝑦 ∘ ^◡𝑦) ⊆ ^◡𝑦)
4		coundir 5637	. . . . . . 7 ⊢ ((^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) = ((^◡𝑦 ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ∪ ((𝑋 ∖ ^◡^◡𝑋) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))))
5		coundi 5636	. . . . . . . . 9 ⊢ (^◡𝑦 ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) = ((^◡𝑦 ∘ ^◡𝑦) ∪ (^◡𝑦 ∘ (𝑋 ∖ ^◡^◡𝑋)))
6		ssid 3624	. . . . . . . . . 10 ⊢ (^◡𝑦 ∘ ^◡𝑦) ⊆ (^◡𝑦 ∘ ^◡𝑦)
7		cononrel2 37901	. . . . . . . . . . 11 ⊢ (^◡𝑦 ∘ (𝑋 ∖ ^◡^◡𝑋)) = ∅
8		0ss 3972	. . . . . . . . . . 11 ⊢ ∅ ⊆ (^◡𝑦 ∘ ^◡𝑦)
9	7, 8	eqsstri 3635	. . . . . . . . . 10 ⊢ (^◡𝑦 ∘ (𝑋 ∖ ^◡^◡𝑋)) ⊆ (^◡𝑦 ∘ ^◡𝑦)
10	6, 9	unssi 3788	. . . . . . . . 9 ⊢ ((^◡𝑦 ∘ ^◡𝑦) ∪ (^◡𝑦 ∘ (𝑋 ∖ ^◡^◡𝑋))) ⊆ (^◡𝑦 ∘ ^◡𝑦)
11	5, 10	eqsstri 3635	. . . . . . . 8 ⊢ (^◡𝑦 ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ⊆ (^◡𝑦 ∘ ^◡𝑦)
12		cononrel1 37900	. . . . . . . . 9 ⊢ ((𝑋 ∖ ^◡^◡𝑋) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) = ∅
13	12, 8	eqsstri 3635	. . . . . . . 8 ⊢ ((𝑋 ∖ ^◡^◡𝑋) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ⊆ (^◡𝑦 ∘ ^◡𝑦)
14	11, 13	unssi 3788	. . . . . . 7 ⊢ ((^◡𝑦 ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ∪ ((𝑋 ∖ ^◡^◡𝑋) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))) ⊆ (^◡𝑦 ∘ ^◡𝑦)
15	4, 14	eqsstri 3635	. . . . . 6 ⊢ ((^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ⊆ (^◡𝑦 ∘ ^◡𝑦)
16		id 22	. . . . . 6 ⊢ ((^◡𝑦 ∘ ^◡𝑦) ⊆ ^◡𝑦 → (^◡𝑦 ∘ ^◡𝑦) ⊆ ^◡𝑦)
17	15, 16	syl5ss 3614	. . . . 5 ⊢ ((^◡𝑦 ∘ ^◡𝑦) ⊆ ^◡𝑦 → ((^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ⊆ ^◡𝑦)
18		ssun3 3778	. . . . 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 5286	. . . . 5 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → (𝑥 ∘ 𝑥) = ((^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))))
22	21, 20	sseq12d 3634	. . . 4 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → ((𝑥 ∘ 𝑥) ⊆ 𝑥 ↔ ((^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ⊆ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))))
23	19, 22	syl5ibr 236	. . 3 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → ((𝑦 ∘ 𝑦) ⊆ 𝑦 → (𝑥 ∘ 𝑥) ⊆ 𝑥))
24	23	adantl 482	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) → ((𝑦 ∘ 𝑦) ⊆ 𝑦 → (𝑥 ∘ 𝑥) ⊆ 𝑥))
25		cnvco 5308	. . . . 5 ⊢ ^◡(𝑥 ∘ 𝑥) = (^◡𝑥 ∘ ^◡𝑥)
26		cnvss 5294	. . . . 5 ⊢ ((𝑥 ∘ 𝑥) ⊆ 𝑥 → ^◡(𝑥 ∘ 𝑥) ⊆ ^◡𝑥)
27	25, 26	syl5eqssr 3650	. . . 4 ⊢ ((𝑥 ∘ 𝑥) ⊆ 𝑥 → (^◡𝑥 ∘ ^◡𝑥) ⊆ ^◡𝑥)
28		id 22	. . . . . 6 ⊢ (𝑦 = ^◡𝑥 → 𝑦 = ^◡𝑥)
29	28, 28	coeq12d 5286	. . . . 5 ⊢ (𝑦 = ^◡𝑥 → (𝑦 ∘ 𝑦) = (^◡𝑥 ∘ ^◡𝑥))
30	29, 28	sseq12d 3634	. . . 4 ⊢ (𝑦 = ^◡𝑥 → ((𝑦 ∘ 𝑦) ⊆ 𝑦 ↔ (^◡𝑥 ∘ ^◡𝑥) ⊆ ^◡𝑥))
31	27, 30	syl5ibr 236	. . 3 ⊢ (𝑦 = ^◡𝑥 → ((𝑥 ∘ 𝑥) ⊆ 𝑥 → (𝑦 ∘ 𝑦) ⊆ 𝑦))
32	31	adantl 482	. 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 5286	. . 3 ⊢ (𝑥 = (𝑋 ∪ (dom 𝑋 × ran 𝑋)) → (𝑥 ∘ 𝑥) = ((𝑋 ∪ (dom 𝑋 × ran 𝑋)) ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))))
35	34, 33	sseq12d 3634	. 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 3776	. . 3 ⊢ 𝑋 ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋))
37	36	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
38		trclexlem 13733	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∪ (dom 𝑋 × ran 𝑋)) ∈ V)
39		coundir 5637	. . . . 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 5636	. . . . . . 7 ⊢ (𝑋 ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))) = ((𝑋 ∘ 𝑋) ∪ (𝑋 ∘ (dom 𝑋 × ran 𝑋)))
41		cossxp 5658	. . . . . . . 8 ⊢ (𝑋 ∘ 𝑋) ⊆ (dom 𝑋 × ran 𝑋)
42		cossxp 5658	. . . . . . . . 9 ⊢ (𝑋 ∘ (dom 𝑋 × ran 𝑋)) ⊆ (dom (dom 𝑋 × ran 𝑋) × ran 𝑋)
43		dmxpss 5565	. . . . . . . . . 10 ⊢ dom (dom 𝑋 × ran 𝑋) ⊆ dom 𝑋
44		xpss1 5228	. . . . . . . . . 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 3612	. . . . . . . 8 ⊢ (𝑋 ∘ (dom 𝑋 × ran 𝑋)) ⊆ (dom 𝑋 × ran 𝑋)
47	41, 46	unssi 3788	. . . . . . 7 ⊢ ((𝑋 ∘ 𝑋) ∪ (𝑋 ∘ (dom 𝑋 × ran 𝑋))) ⊆ (dom 𝑋 × ran 𝑋)
48	40, 47	eqsstri 3635	. . . . . 6 ⊢ (𝑋 ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))) ⊆ (dom 𝑋 × ran 𝑋)
49		coundi 5636	. . . . . . 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 5658	. . . . . . . . 9 ⊢ ((dom 𝑋 × ran 𝑋) ∘ 𝑋) ⊆ (dom 𝑋 × ran (dom 𝑋 × ran 𝑋))
51		rnxpss 5566	. . . . . . . . . 10 ⊢ ran (dom 𝑋 × ran 𝑋) ⊆ ran 𝑋
52		xpss2 5229	. . . . . . . . . 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 3612	. . . . . . . 8 ⊢ ((dom 𝑋 × ran 𝑋) ∘ 𝑋) ⊆ (dom 𝑋 × ran 𝑋)
55		xptrrel 13719	. . . . . . . 8 ⊢ ((dom 𝑋 × ran 𝑋) ∘ (dom 𝑋 × ran 𝑋)) ⊆ (dom 𝑋 × ran 𝑋)
56	54, 55	unssi 3788	. . . . . . 7 ⊢ (((dom 𝑋 × ran 𝑋) ∘ 𝑋) ∪ ((dom 𝑋 × ran 𝑋) ∘ (dom 𝑋 × ran 𝑋))) ⊆ (dom 𝑋 × ran 𝑋)
57	49, 56	eqsstri 3635	. . . . . 6 ⊢ ((dom 𝑋 × ran 𝑋) ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))) ⊆ (dom 𝑋 × ran 𝑋)
58	48, 57	unssi 3788	. . . . 5 ⊢ ((𝑋 ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))) ∪ ((dom 𝑋 × ran 𝑋) ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋)))) ⊆ (dom 𝑋 × ran 𝑋)
59	39, 58	eqsstri 3635	. . . 4 ⊢ ((𝑋 ∪ (dom 𝑋 × ran 𝑋)) ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))) ⊆ (dom 𝑋 × ran 𝑋)
60		ssun2 3777	. . . 4 ⊢ (dom 𝑋 × ran 𝑋) ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋))
61	59, 60	sstri 3612	. . 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 37930	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 384 = wceq 1483 ∈ wcel 1990 {cab 2608 ∖ cdif 3571 ∪ cun 3572 ⊆ wss 3574 ∅c0 3915 ∩ cint 4475 × cxp 5112 ^◡ccnv 5113 dom cdm 5114 ran crn 5115 ∘ ccom 5118

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-iota 5851 df-fun 5890 df-fv 5896 df-1st 7168 df-2nd 7169

This theorem is referenced by: (None)

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