tocyccntz - Mathbox for Thierry Arnoux

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Theorem tocyccntz 33804

Description: All elements of a (finite) set of cycles commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 27-Nov-2023.)

**Hypotheses**
Ref	Expression
tocyccntz.s	⊢ 𝑆 = (SymGrp‘𝐷)
tocyccntz.z	⊢ 𝑍 = (Cntz‘𝑆)
tocyccntz.m	⊢ 𝑀 = (toCyc‘𝐷)
tocyccntz.1	⊢ (𝜑 → 𝐷 ∈ 𝑉)
tocyccntz.2	⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 ran 𝑥)
tocyccntz.a	⊢ (𝜑 → 𝐴 ⊆ dom 𝑀)

**Assertion**
Ref	Expression
tocyccntz	⊢ (𝜑 → (𝑀 “ 𝐴) ⊆ (𝑍‘(𝑀 “ 𝐴)))

Distinct variable groups: 𝑥,𝐴 𝑥,𝑀 𝜑,𝑥 Allowed substitution hints: 𝐷(𝑥) 𝑆(𝑥) 𝑉(𝑥) 𝑍(𝑥)

Proof of Theorem tocyccntz Dummy variables 𝑐 𝑠 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tocyccntz.s	. 2 ⊢ 𝑆 = (SymGrp‘𝐷)
2		eqid 2763	. 2 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		tocyccntz.z	. 2 ⊢ 𝑍 = (Cntz‘𝑆)
4		tocyccntz.1	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
5		tocyccntz.m	. . . 4 ⊢ 𝑀 = (toCyc‘𝐷)
6	5, 1, 2	tocycf 33777	. . 3 ⊢ (𝐷 ∈ 𝑉 → 𝑀:{𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐⟶1-1𝐷}⟶(Base‘𝑆))
7		fimass 6760	. . 3 ⊢ (𝑀:{𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐⟶1-1𝐷}⟶(Base‘𝑆) → (𝑀 “ 𝐴) ⊆ (Base‘𝑆))
8	4, 6, 7	3syl 19	. 2 ⊢ (𝜑 → (𝑀 “ 𝐴) ⊆ (Base‘𝑆))
9		difss 4075	. . . . . . 7 ⊢ (𝐴 ∖ (^◡♯ “ {0, 1})) ⊆ 𝐴
10		tocyccntz.2	. . . . . . 7 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 ran 𝑥)
11		disjss1 5081	. . . . . . 7 ⊢ ((𝐴 ∖ (^◡♯ “ {0, 1})) ⊆ 𝐴 → (Disj 𝑥 ∈ 𝐴 ran 𝑥 → Disj 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))ran 𝑥))
12	9, 10, 11	mpsyl 69	. . . . . 6 ⊢ (𝜑 → Disj 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))ran 𝑥)
13	4	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → 𝐷 ∈ 𝑉)
14		tocyccntz.a	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ dom 𝑀)
15	14	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → 𝐴 ⊆ dom 𝑀)
16		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1})))
17	16	eldifad 4054	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → 𝑥 ∈ 𝐴)
18	15, 17	sseldd 3914	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → 𝑥 ∈ dom 𝑀)
19		fdm 6749	. . . . . . . . . . . . 13 ⊢ (𝑀:{𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐⟶1-1𝐷}⟶(Base‘𝑆) → dom 𝑀 = {𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐⟶1-1𝐷})
20	13, 6, 19	3syl 19	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → dom 𝑀 = {𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐⟶1-1𝐷})
21	18, 20	eleqtrd 2866	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → 𝑥 ∈ {𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐⟶1-1𝐷})
22		id 23	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑥 → 𝑐 = 𝑥)
23		dmeq 5897	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑥 → dom 𝑐 = dom 𝑥)
24		eqidd 2764	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑥 → 𝐷 = 𝐷)
25	22, 23, 24	f1eq123d 6846	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑥 → (𝑐:dom 𝑐⟶1-1𝐷 ↔ 𝑥:dom 𝑥⟶1-1𝐷))
26	25	elrab 3648	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐⟶1-1𝐷} ↔ (𝑥 ∈ Word 𝐷 ∧ 𝑥:dom 𝑥⟶1-1𝐷))
27	21, 26	sylib 221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → (𝑥 ∈ Word 𝐷 ∧ 𝑥:dom 𝑥⟶1-1𝐷))
28	27	simpld 500	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → 𝑥 ∈ Word 𝐷)
29	27	simprd 501	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → 𝑥:dom 𝑥⟶1-1𝐷)
30	16	eldifbd 4055	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → ¬ 𝑥 ∈ (^◡♯ “ {0, 1}))
31		hashgt1 33496	. . . . . . . . . . 11 ⊢ (𝑥 ∈ V → (¬ 𝑥 ∈ (^◡♯ “ {0, 1}) ↔ 1 < (♯‘𝑥)))
32	31	elv 3459	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ (^◡♯ “ {0, 1}) ↔ 1 < (♯‘𝑥))
33	30, 32	sylib 221	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → 1 < (♯‘𝑥))
34	5, 13, 28, 29, 33	cycpmrn 33803	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → ran 𝑥 = dom ((𝑀‘𝑥) ∖ I ))
35	16	fvresd 6936	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥) = (𝑀‘𝑥))
36	35	difeq1d 4065	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → (((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥) ∖ I ) = ((𝑀‘𝑥) ∖ I ))
37	36	dmeqd 5899	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → dom (((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥) ∖ I ) = dom ((𝑀‘𝑥) ∖ I ))
38	34, 37	eqtr4d 2801	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → ran 𝑥 = dom (((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥) ∖ I ))
39	38	disjeq2dv 5080	. . . . . 6 ⊢ (𝜑 → (Disj 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))ran 𝑥 ↔ Disj 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))dom (((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥) ∖ I )))
40	12, 39	mpbid 235	. . . . 5 ⊢ (𝜑 → Disj 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))dom (((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥) ∖ I ))
41	4, 6	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀:{𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐⟶1-1𝐷}⟶(Base‘𝑆))
42	41	ffdmd 6770	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀:dom 𝑀⟶(Base‘𝑆))
43	14	ssdifssd 4086	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∖ (^◡♯ “ {0, 1})) ⊆ dom 𝑀)
44	42, 43	fssresd 6779	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))):(𝐴 ∖ (^◡♯ “ {0, 1}))⟶(Base‘𝑆))
45	41, 14	fssdmd 6758	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐴 ⊆ {𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐⟶1-1𝐷})
46	45	ad4antr 745	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝐴 ⊆ {𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐⟶1-1𝐷})
47		simp-4r 796	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1})))
48	47	eldifad 4054	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠 ∈ 𝐴)
49	46, 48	sseldd 3914	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠 ∈ {𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐⟶1-1𝐷})
50		id 23	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = 𝑠 → 𝑐 = 𝑠)
51		dmeq 5897	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = 𝑠 → dom 𝑐 = dom 𝑠)
52		eqidd 2764	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = 𝑠 → 𝐷 = 𝐷)
53	50, 51, 52	f1eq123d 6846	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = 𝑠 → (𝑐:dom 𝑐⟶1-1𝐷 ↔ 𝑠:dom 𝑠⟶1-1𝐷))
54	53	elrab 3648	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ {𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐⟶1-1𝐷} ↔ (𝑠 ∈ Word 𝐷 ∧ 𝑠:dom 𝑠⟶1-1𝐷))
55	49, 54	sylib 221	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → (𝑠 ∈ Word 𝐷 ∧ 𝑠:dom 𝑠⟶1-1𝐷))
56	55	simpld 500	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠 ∈ Word 𝐷)
57		wrdf 14704	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ Word 𝐷 → 𝑠:(0..^(♯‘𝑠))⟶𝐷)
58		frel 6745	. . . . . . . . . . . . . . . 16 ⊢ (𝑠:(0..^(♯‘𝑠))⟶𝐷 → Rel 𝑠)
59	56, 57, 58	3syl 19	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → Rel 𝑠)
60		simplr 781	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥))
61	47	fvresd 6936	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = (𝑀‘𝑠))
62	16	ad5ant13 769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1})))
63	62	fvresd 6936	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥) = (𝑀‘𝑥))
64	60, 61, 63	3eqtr3rd 2807	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → (𝑀‘𝑥) = (𝑀‘𝑠))
65	64	difeq1d 4065	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ((𝑀‘𝑥) ∖ I ) = ((𝑀‘𝑠) ∖ I ))
66	65	dmeqd 5899	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → dom ((𝑀‘𝑥) ∖ I ) = dom ((𝑀‘𝑠) ∖ I ))
67	4	ad4antr 745	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝐷 ∈ 𝑉)
68	17	ad5ant13 769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑥 ∈ 𝐴)
69	46, 68	sseldd 3914	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑥 ∈ {𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐⟶1-1𝐷})
70	69, 26	sylib 221	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → (𝑥 ∈ Word 𝐷 ∧ 𝑥:dom 𝑥⟶1-1𝐷))
71	70	simpld 500	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑥 ∈ Word 𝐷)
72	70	simprd 501	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑥:dom 𝑥⟶1-1𝐷)
73	33	ad5ant13 769	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 1 < (♯‘𝑥))
74	5, 67, 71, 72, 73	cycpmrn 33803	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ran 𝑥 = dom ((𝑀‘𝑥) ∖ I ))
75	55	simprd 501	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠:dom 𝑠⟶1-1𝐷)
76	14	ssdifd 4084	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐴 ∖ (^◡♯ “ {0, 1})) ⊆ (dom 𝑀 ∖ (^◡♯ “ {0, 1})))
77	76	sselda 3913	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → 𝑠 ∈ (dom 𝑀 ∖ (^◡♯ “ {0, 1})))
78	77	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠 ∈ (dom 𝑀 ∖ (^◡♯ “ {0, 1})))
79	78	eldifbd 4055	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ¬ 𝑠 ∈ (^◡♯ “ {0, 1}))
80		hashgt1 33496	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 ∈ 𝐴 → (¬ 𝑠 ∈ (^◡♯ “ {0, 1}) ↔ 1 < (♯‘𝑠)))
81	80	biimpa 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ∈ (^◡♯ “ {0, 1})) → 1 < (♯‘𝑠))
82	48, 79, 81	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 1 < (♯‘𝑠))
83	5, 67, 56, 75, 82	cycpmrn 33803	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ran 𝑠 = dom ((𝑀‘𝑠) ∖ I ))
84	66, 74, 83	3eqtr4rd 2809	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ran 𝑠 = ran 𝑥)
85	84	ineq2d 4120	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → (ran 𝑥 ∩ ran 𝑠) = (ran 𝑥 ∩ ran 𝑥))
86		inidm 4126	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝑥 ∩ ran 𝑥) = ran 𝑥
87	85, 86	eqtrdi 2814	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → (ran 𝑥 ∩ ran 𝑠) = ran 𝑥)
88		rneq 5930	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑦 → ran 𝑥 = ran 𝑦)
89	88	cbvdisjv 5086	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Disj 𝑥 ∈ 𝐴 ran 𝑥 ↔ Disj 𝑦 ∈ 𝐴 ran 𝑦)
90	10, 89	sylib 221	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Disj 𝑦 ∈ 𝐴 ran 𝑦)
91	90	ad4antr 745	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → Disj 𝑦 ∈ 𝐴 ran 𝑦)
92		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ¬ 𝑠 = 𝑥)
93	92	neqned 2966	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠 ≠ 𝑥)
94	93	necomd 3014	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑥 ≠ 𝑠)
95		rneq 5930	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑥 → ran 𝑦 = ran 𝑥)
96		rneq 5930	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑠 → ran 𝑦 = ran 𝑠)
97	95, 96	disji2 5092	. . . . . . . . . . . . . . . . . 18 ⊢ ((Disj 𝑦 ∈ 𝐴 ran 𝑦 ∧ (𝑥 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ 𝑥 ≠ 𝑠) → (ran 𝑥 ∩ ran 𝑠) = ∅)
98	91, 68, 48, 94, 97	syl121anc 1403	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → (ran 𝑥 ∩ ran 𝑠) = ∅)
99	87, 98	eqtr3d 2800	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ran 𝑥 = ∅)
100	84, 99	eqtrd 2798	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ran 𝑠 = ∅)
101		relrn0 5967	. . . . . . . . . . . . . . . 16 ⊢ (Rel 𝑠 → (𝑠 = ∅ ↔ ran 𝑠 = ∅))
102	101	biimpar 483	. . . . . . . . . . . . . . 15 ⊢ ((Rel 𝑠 ∧ ran 𝑠 = ∅) → 𝑠 = ∅)
103	59, 100, 102	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠 = ∅)
104		wrdf 14704	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ Word 𝐷 → 𝑥:(0..^(♯‘𝑥))⟶𝐷)
105		frel 6745	. . . . . . . . . . . . . . . 16 ⊢ (𝑥:(0..^(♯‘𝑥))⟶𝐷 → Rel 𝑥)
106	71, 104, 105	3syl 19	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → Rel 𝑥)
107		relrn0 5967	. . . . . . . . . . . . . . . 16 ⊢ (Rel 𝑥 → (𝑥 = ∅ ↔ ran 𝑥 = ∅))
108	107	biimpar 483	. . . . . . . . . . . . . . 15 ⊢ ((Rel 𝑥 ∧ ran 𝑥 = ∅) → 𝑥 = ∅)
109	106, 99, 108	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑥 = ∅)
110	103, 109	eqtr4d 2801	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠 = 𝑥)
111	110	pm2.18da 812	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) → 𝑠 = 𝑥)
112	111	ex 418	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → (((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥) → 𝑠 = 𝑥))
113	112	anasss 472	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1})) ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1})))) → (((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥) → 𝑠 = 𝑥))
114	113	ralrimivva 3209	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))∀𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))(((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥) → 𝑠 = 𝑥))
115		dff13 7289	. . . . . . . . 9 ⊢ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))):(𝐴 ∖ (^◡♯ “ {0, 1}))⟶1-1(Base‘𝑆) ↔ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))):(𝐴 ∖ (^◡♯ “ {0, 1}))⟶(Base‘𝑆) ∧ ∀𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))∀𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))(((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥) → 𝑠 = 𝑥)))
116	44, 114, 115	sylanbrc 595	. . . . . . . 8 ⊢ (𝜑 → (𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))):(𝐴 ∖ (^◡♯ “ {0, 1}))⟶1-1(Base‘𝑆))
117		f1f1orn 6866	. . . . . . . 8 ⊢ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))):(𝐴 ∖ (^◡♯ “ {0, 1}))⟶1-1(Base‘𝑆) → (𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))):(𝐴 ∖ (^◡♯ “ {0, 1}))⟶1-1ontoran (𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))))
118	116, 117	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))):(𝐴 ∖ (^◡♯ “ {0, 1}))⟶1-1ontoran (𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))))
119		df-ima 5671	. . . . . . . . 9 ⊢ (𝑀 “ (𝐴 ∖ (^◡♯ “ {0, 1}))) = ran (𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))
120	119	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝑀 “ (𝐴 ∖ (^◡♯ “ {0, 1}))) = ran (𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))))
121	120	f1oeq3d 6851	. . . . . . 7 ⊢ (𝜑 → ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))):(𝐴 ∖ (^◡♯ “ {0, 1}))⟶1-1onto(𝑀 “ (𝐴 ∖ (^◡♯ “ {0, 1}))) ↔ (𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))):(𝐴 ∖ (^◡♯ “ {0, 1}))⟶1-1ontoran (𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))))
122	118, 121	mpbird 260	. . . . . 6 ⊢ (𝜑 → (𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))):(𝐴 ∖ (^◡♯ “ {0, 1}))⟶1-1onto(𝑀 “ (𝐴 ∖ (^◡♯ “ {0, 1}))))
123		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) → 𝑐 = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥))
124	123	difeq1d 4065	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) → (𝑐 ∖ I ) = (((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥) ∖ I ))
125	124	dmeqd 5899	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) → dom (𝑐 ∖ I ) = dom (((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥) ∖ I ))
126	122, 125	disjrdx 33283	. . . . 5 ⊢ (𝜑 → (Disj 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))dom (((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥) ∖ I ) ↔ Disj 𝑐 ∈ (𝑀 “ (𝐴 ∖ (^◡♯ “ {0, 1})))dom (𝑐 ∖ I )))
127	40, 126	mpbid 235	. . . 4 ⊢ (𝜑 → Disj 𝑐 ∈ (𝑀 “ (𝐴 ∖ (^◡♯ “ {0, 1})))dom (𝑐 ∖ I ))
128		simpr 490	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∧ (𝑀‘𝑥) = 𝑐) → (𝑀‘𝑥) = 𝑐)
129	4	ad3antrrr 743	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∧ (𝑀‘𝑥) = 𝑐) → 𝐷 ∈ 𝑉)
130	14	ssrind 4143	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∩ (^◡♯ “ {0, 1})) ⊆ (dom 𝑀 ∩ (^◡♯ “ {0, 1})))
131	130	ad3antrrr 743	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∧ (𝑀‘𝑥) = 𝑐) → (𝐴 ∩ (^◡♯ “ {0, 1})) ⊆ (dom 𝑀 ∩ (^◡♯ “ {0, 1})))
132		simplr 781	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∧ (𝑀‘𝑥) = 𝑐) → 𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1})))
133	131, 132	sseldd 3914	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∧ (𝑀‘𝑥) = 𝑐) → 𝑥 ∈ (dom 𝑀 ∩ (^◡♯ “ {0, 1})))
134	5	tocyc01 33778	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑥 ∈ (dom 𝑀 ∩ (^◡♯ “ {0, 1}))) → (𝑀‘𝑥) = ( I ↾ 𝐷))
135	129, 133, 134	syl2anc 596	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∧ (𝑀‘𝑥) = 𝑐) → (𝑀‘𝑥) = ( I ↾ 𝐷))
136	128, 135	eqtr3d 2800	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∧ (𝑀‘𝑥) = 𝑐) → 𝑐 = ( I ↾ 𝐷))
137	136	difeq1d 4065	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∧ (𝑀‘𝑥) = 𝑐) → (𝑐 ∖ I ) = (( I ↾ 𝐷) ∖ I ))
138	137	dmeqd 5899	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∧ (𝑀‘𝑥) = 𝑐) → dom (𝑐 ∖ I ) = dom (( I ↾ 𝐷) ∖ I ))
139		resdifcom 6005	. . . . . . . . . 10 ⊢ (( I ↾ 𝐷) ∖ I ) = (( I ∖ I ) ↾ 𝐷)
140		difid 4328	. . . . . . . . . . 11 ⊢ ( I ∖ I ) = ∅
141	140	reseq1i 5982	. . . . . . . . . 10 ⊢ (( I ∖ I ) ↾ 𝐷) = (∅ ↾ 𝐷)
142		0res 33295	. . . . . . . . . 10 ⊢ (∅ ↾ 𝐷) = ∅
143	139, 141, 142	3eqtri 2790	. . . . . . . . 9 ⊢ (( I ↾ 𝐷) ∖ I ) = ∅
144	143	dmeqi 5898	. . . . . . . 8 ⊢ dom (( I ↾ 𝐷) ∖ I ) = dom ∅
145		dm0 5914	. . . . . . . 8 ⊢ dom ∅ = ∅
146	144, 145	eqtri 2786	. . . . . . 7 ⊢ dom (( I ↾ 𝐷) ∖ I ) = ∅
147	138, 146	eqtrdi 2814	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∧ (𝑀‘𝑥) = 𝑐) → dom (𝑐 ∖ I ) = ∅)
148	41	ffund 6744	. . . . . . 7 ⊢ (𝜑 → Fun 𝑀)
149		fvelima 6981	. . . . . . 7 ⊢ ((Fun 𝑀 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) → ∃𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))(𝑀‘𝑥) = 𝑐)
150	148, 149	sylan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) → ∃𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))(𝑀‘𝑥) = 𝑐)
151	147, 150	r19.29a 3174	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) → dom (𝑐 ∖ I ) = ∅)
152	151	disjxun0 33266	. . . 4 ⊢ (𝜑 → (Disj 𝑐 ∈ ((𝑀 “ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1}))))dom (𝑐 ∖ I ) ↔ Disj 𝑐 ∈ (𝑀 “ (𝐴 ∖ (^◡♯ “ {0, 1})))dom (𝑐 ∖ I )))
153	127, 152	mpbird 260	. . 3 ⊢ (𝜑 → Disj 𝑐 ∈ ((𝑀 “ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1}))))dom (𝑐 ∖ I ))
154		uncom 4160	. . . . . 6 ⊢ ((𝑀 “ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) = ((𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∖ (^◡♯ “ {0, 1}))))
155		imaundi 6159	. . . . . 6 ⊢ (𝑀 “ ((𝐴 ∩ (^◡♯ “ {0, 1})) ∪ (𝐴 ∖ (^◡♯ “ {0, 1})))) = ((𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∖ (^◡♯ “ {0, 1}))))
156		inundif 4438	. . . . . . 7 ⊢ ((𝐴 ∩ (^◡♯ “ {0, 1})) ∪ (𝐴 ∖ (^◡♯ “ {0, 1}))) = 𝐴
157	156	imaeq2i 6068	. . . . . 6 ⊢ (𝑀 “ ((𝐴 ∩ (^◡♯ “ {0, 1})) ∪ (𝐴 ∖ (^◡♯ “ {0, 1})))) = (𝑀 “ 𝐴)
158	154, 155, 157	3eqtr2i 2792	. . . . 5 ⊢ ((𝑀 “ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) = (𝑀 “ 𝐴)
159	158	a1i 11	. . . 4 ⊢ (𝜑 → ((𝑀 “ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) = (𝑀 “ 𝐴))
160	159	disjeq1d 5083	. . 3 ⊢ (𝜑 → (Disj 𝑐 ∈ ((𝑀 “ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1}))))dom (𝑐 ∖ I ) ↔ Disj 𝑐 ∈ (𝑀 “ 𝐴)dom (𝑐 ∖ I )))
161	153, 160	mpbid 235	. 2 ⊢ (𝜑 → Disj 𝑐 ∈ (𝑀 “ 𝐴)dom (𝑐 ∖ I ))
162	1, 2, 3, 8, 161	symgcntz 33745	1 ⊢ (𝜑 → (𝑀 “ 𝐴) ⊆ (𝑍‘(𝑀 “ 𝐴)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 ∃wrex 3090 {crab 3416 Vcvv 3454 ⊆ wss 3900 ∖ cdif 4048 ∩ cin 4092 ∪ cun 4152 ∅c0 4281 {cpr 4590 Disj wdisj 5075 class class class wbr 5108 I cid 5552 ^◡ccnv 5657 dom cdm 5658 ran crn 5659 ↾ cres 5660 “ cima 5661 Rel wrel 5663 Fun wfun 6560 ⟶wf 6562 ⟶1-1wf1 6563 ⟶1-1ontowf1o 6565 ‘cfv 6566 (class class class)co 7451 0cc0 11239 1c1 11240 < clt 11382 ..^cfzo 13828 ♯chash 14515 Word cword 14699 Basecbs 17431 Cntzccntz 19580 SymGrpcsymg 19634 toCycctocyc 33766

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-disj 5076 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-er 8755 df-map 8887 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-sup 9471 df-inf 9472 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-xnn0 12720 df-z 12734 df-uz 13006 df-rp 13161 df-fz 13680 df-fzo 13829 df-fl 13972 df-mod 14050 df-hash 14516 df-word 14700 df-concat 14757 df-substr 14831 df-pfx 14863 df-csh 14982 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-tset 17492 df-efmnd 19115 df-cntz 19582 df-symg 19635 df-tocyc 33767

This theorem is used by: (None)

W3C validator