cnvtrcl0 - Mathbox for Richard Penner

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Theorem cnvtrcl0 44570

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 5863	. . . . . 6 ⊢ ^◡(𝑦 ∘ 𝑦) = (^◡𝑦 ∘ ^◡𝑦)
2		cnvss 5846	. . . . . 6 ⊢ ((𝑦 ∘ 𝑦) ⊆ 𝑦 → ^◡(𝑦 ∘ 𝑦) ⊆ ^◡𝑦)
3	1, 2	eqsstrrid 3969	. . . . 5 ⊢ ((𝑦 ∘ 𝑦) ⊆ 𝑦 → (^◡𝑦 ∘ ^◡𝑦) ⊆ ^◡𝑦)
4		coundir 6238	. . . . . . 7 ⊢ ((^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) = ((^◡𝑦 ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ∪ ((𝑋 ∖ ^◡^◡𝑋) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))))
5		coundi 6237	. . . . . . . . 9 ⊢ (^◡𝑦 ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) = ((^◡𝑦 ∘ ^◡𝑦) ∪ (^◡𝑦 ∘ (𝑋 ∖ ^◡^◡𝑋)))
6		ssid 3952	. . . . . . . . . 10 ⊢ (^◡𝑦 ∘ ^◡𝑦) ⊆ (^◡𝑦 ∘ ^◡𝑦)
7		cononrel2 44539	. . . . . . . . . . 11 ⊢ (^◡𝑦 ∘ (𝑋 ∖ ^◡^◡𝑋)) = ∅
8		0ss 4349	. . . . . . . . . . 11 ⊢ ∅ ⊆ (^◡𝑦 ∘ ^◡𝑦)
9	7, 8	eqsstri 3976	. . . . . . . . . 10 ⊢ (^◡𝑦 ∘ (𝑋 ∖ ^◡^◡𝑋)) ⊆ (^◡𝑦 ∘ ^◡𝑦)
10	6, 9	unssi 4136	. . . . . . . . 9 ⊢ ((^◡𝑦 ∘ ^◡𝑦) ∪ (^◡𝑦 ∘ (𝑋 ∖ ^◡^◡𝑋))) ⊆ (^◡𝑦 ∘ ^◡𝑦)
11	5, 10	eqsstri 3976	. . . . . . . 8 ⊢ (^◡𝑦 ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ⊆ (^◡𝑦 ∘ ^◡𝑦)
12		cononrel1 44538	. . . . . . . . 9 ⊢ ((𝑋 ∖ ^◡^◡𝑋) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) = ∅
13	12, 8	eqsstri 3976	. . . . . . . 8 ⊢ ((𝑋 ∖ ^◡^◡𝑋) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ⊆ (^◡𝑦 ∘ ^◡𝑦)
14	11, 13	unssi 4136	. . . . . . 7 ⊢ ((^◡𝑦 ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ∪ ((𝑋 ∖ ^◡^◡𝑋) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))) ⊆ (^◡𝑦 ∘ ^◡𝑦)
15	4, 14	eqsstri 3976	. . . . . 6 ⊢ ((^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ⊆ (^◡𝑦 ∘ ^◡𝑦)
16		id 23	. . . . . 6 ⊢ ((^◡𝑦 ∘ ^◡𝑦) ⊆ ^◡𝑦 → (^◡𝑦 ∘ ^◡𝑦) ⊆ ^◡𝑦)
17	15, 16	sstrid 3941	. . . . 5 ⊢ ((^◡𝑦 ∘ ^◡𝑦) ⊆ ^◡𝑦 → ((^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ⊆ ^◡𝑦)
18		ssun3 4125	. . . . 5 ⊢ (((^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ⊆ ^◡𝑦 → ((^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ⊆ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
19	3, 17, 18	3syl 19	. . . 4 ⊢ ((𝑦 ∘ 𝑦) ⊆ 𝑦 → ((^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ⊆ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
20		id 23	. . . . . 6 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → 𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)))
21	20, 20	coeq12d 5838	. . . . 5 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → (𝑥 ∘ 𝑥) = ((^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))))
22	21, 20	sseq12d 3963	. . . 4 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → ((𝑥 ∘ 𝑥) ⊆ 𝑥 ↔ ((^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) ∘ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) ⊆ (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))))
23	19, 22	imbitrrid 249	. . 3 ⊢ (𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋)) → ((𝑦 ∘ 𝑦) ⊆ 𝑦 → (𝑥 ∘ 𝑥) ⊆ 𝑥))
24	23	adantl 487	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = (^◡𝑦 ∪ (𝑋 ∖ ^◡^◡𝑋))) → ((𝑦 ∘ 𝑦) ⊆ 𝑦 → (𝑥 ∘ 𝑥) ⊆ 𝑥))
25		cnvco 5863	. . . . 5 ⊢ ^◡(𝑥 ∘ 𝑥) = (^◡𝑥 ∘ ^◡𝑥)
26		cnvss 5846	. . . . 5 ⊢ ((𝑥 ∘ 𝑥) ⊆ 𝑥 → ^◡(𝑥 ∘ 𝑥) ⊆ ^◡𝑥)
27	25, 26	eqsstrrid 3969	. . . 4 ⊢ ((𝑥 ∘ 𝑥) ⊆ 𝑥 → (^◡𝑥 ∘ ^◡𝑥) ⊆ ^◡𝑥)
28		id 23	. . . . . 6 ⊢ (𝑦 = ^◡𝑥 → 𝑦 = ^◡𝑥)
29	28, 28	coeq12d 5838	. . . . 5 ⊢ (𝑦 = ^◡𝑥 → (𝑦 ∘ 𝑦) = (^◡𝑥 ∘ ^◡𝑥))
30	29, 28	sseq12d 3963	. . . 4 ⊢ (𝑦 = ^◡𝑥 → ((𝑦 ∘ 𝑦) ⊆ 𝑦 ↔ (^◡𝑥 ∘ ^◡𝑥) ⊆ ^◡𝑥))
31	27, 30	imbitrrid 249	. . 3 ⊢ (𝑦 = ^◡𝑥 → ((𝑥 ∘ 𝑥) ⊆ 𝑥 → (𝑦 ∘ 𝑦) ⊆ 𝑦))
32	31	adantl 487	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 = ^◡𝑥) → ((𝑥 ∘ 𝑥) ⊆ 𝑥 → (𝑦 ∘ 𝑦) ⊆ 𝑦))
33		id 23	. . . 4 ⊢ (𝑥 = (𝑋 ∪ (dom 𝑋 × ran 𝑋)) → 𝑥 = (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
34	33, 33	coeq12d 5838	. . 3 ⊢ (𝑥 = (𝑋 ∪ (dom 𝑋 × ran 𝑋)) → (𝑥 ∘ 𝑥) = ((𝑋 ∪ (dom 𝑋 × ran 𝑋)) ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))))
35	34, 33	sseq12d 3963	. 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 4123	. . 3 ⊢ 𝑋 ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋))
37	36	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
38		trclexlem 15115	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∪ (dom 𝑋 × ran 𝑋)) ∈ V)
39		coundir 6238	. . . . 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 6237	. . . . . . 7 ⊢ (𝑋 ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))) = ((𝑋 ∘ 𝑋) ∪ (𝑋 ∘ (dom 𝑋 × ran 𝑋)))
41		cossxp 6263	. . . . . . . 8 ⊢ (𝑋 ∘ 𝑋) ⊆ (dom 𝑋 × ran 𝑋)
42		cossxp 6263	. . . . . . . . 9 ⊢ (𝑋 ∘ (dom 𝑋 × ran 𝑋)) ⊆ (dom (dom 𝑋 × ran 𝑋) × ran 𝑋)
43		dmxpss 6158	. . . . . . . . . 10 ⊢ dom (dom 𝑋 × ran 𝑋) ⊆ dom 𝑋
44		xpss1 5666	. . . . . . . . . 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 3939	. . . . . . . 8 ⊢ (𝑋 ∘ (dom 𝑋 × ran 𝑋)) ⊆ (dom 𝑋 × ran 𝑋)
47	41, 46	unssi 4136	. . . . . . 7 ⊢ ((𝑋 ∘ 𝑋) ∪ (𝑋 ∘ (dom 𝑋 × ran 𝑋))) ⊆ (dom 𝑋 × ran 𝑋)
48	40, 47	eqsstri 3976	. . . . . 6 ⊢ (𝑋 ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))) ⊆ (dom 𝑋 × ran 𝑋)
49		coundi 6237	. . . . . . 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 6263	. . . . . . . . 9 ⊢ ((dom 𝑋 × ran 𝑋) ∘ 𝑋) ⊆ (dom 𝑋 × ran (dom 𝑋 × ran 𝑋))
51		rnxpss 6159	. . . . . . . . . 10 ⊢ ran (dom 𝑋 × ran 𝑋) ⊆ ran 𝑋
52		xpss2 5667	. . . . . . . . . 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 3939	. . . . . . . 8 ⊢ ((dom 𝑋 × ran 𝑋) ∘ 𝑋) ⊆ (dom 𝑋 × ran 𝑋)
55		xptrrel 15101	. . . . . . . 8 ⊢ ((dom 𝑋 × ran 𝑋) ∘ (dom 𝑋 × ran 𝑋)) ⊆ (dom 𝑋 × ran 𝑋)
56	54, 55	unssi 4136	. . . . . . 7 ⊢ (((dom 𝑋 × ran 𝑋) ∘ 𝑋) ∪ ((dom 𝑋 × ran 𝑋) ∘ (dom 𝑋 × ran 𝑋))) ⊆ (dom 𝑋 × ran 𝑋)
57	49, 56	eqsstri 3976	. . . . . 6 ⊢ ((dom 𝑋 × ran 𝑋) ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))) ⊆ (dom 𝑋 × ran 𝑋)
58	48, 57	unssi 4136	. . . . 5 ⊢ ((𝑋 ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))) ∪ ((dom 𝑋 × ran 𝑋) ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋)))) ⊆ (dom 𝑋 × ran 𝑋)
59	39, 58	eqsstri 3976	. . . 4 ⊢ ((𝑋 ∪ (dom 𝑋 × ran 𝑋)) ∘ (𝑋 ∪ (dom 𝑋 × ran 𝑋))) ⊆ (dom 𝑋 × ran 𝑋)
60		ssun2 4124	. . . 4 ⊢ (dom 𝑋 × ran 𝑋) ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋))
61	59, 60	sstri 3939	. . 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 44567	1 ⊢ (𝑋 ∈ 𝑉 → ^◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = ∩ {𝑦 ∣ (^◡𝑋 ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦)})

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 {cab 2738 ∖ cdif 3895 ∪ cun 3896 ⊆ wss 3898 ∅c0 4278 ∩ cint 4906 × cxp 5645 ^◡ccnv 5646 dom cdm 5647 ran crn 5648 ∘ ccom 5651

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5542 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-iota 6483 df-fun 6529 df-fv 6535 df-1st 7984 df-2nd 7985

This theorem is used by: (None)

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