tocyccntz - Mathbox for Thierry Arnoux

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Theorem tocyccntz 33160

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 2736	. 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 33133	. . 3 ⊢ (𝐷 ∈ 𝑉 → 𝑀:{𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐–1-1→𝐷}⟶(Base‘𝑆))
7		fimass 6731	. . 3 ⊢ (𝑀:{𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐–1-1→𝐷}⟶(Base‘𝑆) → (𝑀 “ 𝐴) ⊆ (Base‘𝑆))
8	4, 6, 7	3syl 18	. 2 ⊢ (𝜑 → (𝑀 “ 𝐴) ⊆ (Base‘𝑆))
9		difss 4116	. . . . . . 7 ⊢ (𝐴 ∖ (^◡♯ “ {0, 1})) ⊆ 𝐴
10		tocyccntz.2	. . . . . . 7 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 ran 𝑥)
11		disjss1 5097	. . . . . . 7 ⊢ ((𝐴 ∖ (^◡♯ “ {0, 1})) ⊆ 𝐴 → (Disj 𝑥 ∈ 𝐴 ran 𝑥 → Disj 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))ran 𝑥))
12	9, 10, 11	mpsyl 68	. . . . . 6 ⊢ (𝜑 → Disj 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))ran 𝑥)
13	4	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → 𝐷 ∈ 𝑉)
14		tocyccntz.a	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ dom 𝑀)
15	14	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → 𝐴 ⊆ dom 𝑀)
16		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1})))
17	16	eldifad 3943	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → 𝑥 ∈ 𝐴)
18	15, 17	sseldd 3964	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → 𝑥 ∈ dom 𝑀)
19		fdm 6720	. . . . . . . . . . . . 13 ⊢ (𝑀:{𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐–1-1→𝐷}⟶(Base‘𝑆) → dom 𝑀 = {𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐–1-1→𝐷})
20	13, 6, 19	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → dom 𝑀 = {𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐–1-1→𝐷})
21	18, 20	eleqtrd 2837	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → 𝑥 ∈ {𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐–1-1→𝐷})
22		id 22	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑥 → 𝑐 = 𝑥)
23		dmeq 5888	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑥 → dom 𝑐 = dom 𝑥)
24		eqidd 2737	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑥 → 𝐷 = 𝐷)
25	22, 23, 24	f1eq123d 6815	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑥 → (𝑐:dom 𝑐–1-1→𝐷 ↔ 𝑥:dom 𝑥–1-1→𝐷))
26	25	elrab 3676	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐–1-1→𝐷} ↔ (𝑥 ∈ Word 𝐷 ∧ 𝑥:dom 𝑥–1-1→𝐷))
27	21, 26	sylib 218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → (𝑥 ∈ Word 𝐷 ∧ 𝑥:dom 𝑥–1-1→𝐷))
28	27	simpld 494	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → 𝑥 ∈ Word 𝐷)
29	27	simprd 495	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → 𝑥:dom 𝑥–1-1→𝐷)
30	16	eldifbd 3944	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → ¬ 𝑥 ∈ (^◡♯ “ {0, 1}))
31		hashgt1 32792	. . . . . . . . . . 11 ⊢ (𝑥 ∈ V → (¬ 𝑥 ∈ (^◡♯ “ {0, 1}) ↔ 1 < (♯‘𝑥)))
32	31	elv 3469	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ (^◡♯ “ {0, 1}) ↔ 1 < (♯‘𝑥))
33	30, 32	sylib 218	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → 1 < (♯‘𝑥))
34	5, 13, 28, 29, 33	cycpmrn 33159	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → ran 𝑥 = dom ((𝑀‘𝑥) ∖ I ))
35	16	fvresd 6901	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥) = (𝑀‘𝑥))
36	35	difeq1d 4105	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → (((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥) ∖ I ) = ((𝑀‘𝑥) ∖ I ))
37	36	dmeqd 5890	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → dom (((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥) ∖ I ) = dom ((𝑀‘𝑥) ∖ I ))
38	34, 37	eqtr4d 2774	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → ran 𝑥 = dom (((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥) ∖ I ))
39	38	disjeq2dv 5096	. . . . . 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 232	. . . . 5 ⊢ (𝜑 → Disj 𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))dom (((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥) ∖ I ))
41	4, 6	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀:{𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐–1-1→𝐷}⟶(Base‘𝑆))
42	41	ffdmd 6741	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀:dom 𝑀⟶(Base‘𝑆))
43	14	ssdifssd 4127	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∖ (^◡♯ “ {0, 1})) ⊆ dom 𝑀)
44	42, 43	fssresd 6750	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))):(𝐴 ∖ (^◡♯ “ {0, 1}))⟶(Base‘𝑆))
45	41, 14	fssdmd 6729	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐴 ⊆ {𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐–1-1→𝐷})
46	45	ad4antr 732	. . . . . . . . . . . . . . . . . . 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–1-1→𝐷})
47		simp-4r 783	. . . . . . . . . . . . . . . . . . . 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 3943	. . . . . . . . . . . . . . . . . . 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 3964	. . . . . . . . . . . . . . . . . 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–1-1→𝐷})
50		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = 𝑠 → 𝑐 = 𝑠)
51		dmeq 5888	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = 𝑠 → dom 𝑐 = dom 𝑠)
52		eqidd 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = 𝑠 → 𝐷 = 𝐷)
53	50, 51, 52	f1eq123d 6815	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = 𝑠 → (𝑐:dom 𝑐–1-1→𝐷 ↔ 𝑠:dom 𝑠–1-1→𝐷))
54	53	elrab 3676	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ {𝑐 ∈ Word 𝐷 ∣ 𝑐:dom 𝑐–1-1→𝐷} ↔ (𝑠 ∈ Word 𝐷 ∧ 𝑠:dom 𝑠–1-1→𝐷))
55	49, 54	sylib 218	. . . . . . . . . . . . . . . . 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–1-1→𝐷))
56	55	simpld 494	. . . . . . . . . . . . . . . 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 14541	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ Word 𝐷 → 𝑠:(0..^(♯‘𝑠))⟶𝐷)
58		frel 6716	. . . . . . . . . . . . . . . 16 ⊢ (𝑠:(0..^(♯‘𝑠))⟶𝐷 → Rel 𝑠)
59	56, 57, 58	3syl 18	. . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . . . . 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 6901	. . . . . . . . . . . . . . . . . . . 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 756	. . . . . . . . . . . . . . . . . . . . 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 6901	. . . . . . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . . . . . 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 4105	. . . . . . . . . . . . . . . . . 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 5890	. . . . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . . . . 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 756	. . . . . . . . . . . . . . . . . . . . 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 3964	. . . . . . . . . . . . . . . . . . . 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–1-1→𝐷})
70	69, 26	sylib 218	. . . . . . . . . . . . . . . . . . 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–1-1→𝐷))
71	70	simpld 494	. . . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . . . 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–1-1→𝐷)
73	33	ad5ant13 756	. . . . . . . . . . . . . . . . . 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 33159	. . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . . . 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–1-1→𝐷)
76	14	ssdifd 4125	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐴 ∖ (^◡♯ “ {0, 1})) ⊆ (dom 𝑀 ∖ (^◡♯ “ {0, 1})))
77	76	sselda 3963	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))) → 𝑠 ∈ (dom 𝑀 ∖ (^◡♯ “ {0, 1})))
78	77	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . . . . . . . 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 32792	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 ∈ 𝐴 → (¬ 𝑠 ∈ (^◡♯ “ {0, 1}) ↔ 1 < (♯‘𝑠)))
81	80	biimpa 476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ∈ (^◡♯ “ {0, 1})) → 1 < (♯‘𝑠))
82	48, 79, 81	syl2anc 584	. . . . . . . . . . . . . . . . . 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 33159	. . . . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . . . 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 4200	. . . . . . . . . . . . . . . . . 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 4207	. . . . . . . . . . . . . . . . . 18 ⊢ (ran 𝑥 ∩ ran 𝑥) = ran 𝑥
87	85, 86	eqtrdi 2787	. . . . . . . . . . . . . . . . 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 5921	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑦 → ran 𝑥 = ran 𝑦)
89	88	cbvdisjv 5102	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Disj 𝑥 ∈ 𝐴 ran 𝑥 ↔ Disj 𝑦 ∈ 𝐴 ran 𝑦)
90	10, 89	sylib 218	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Disj 𝑦 ∈ 𝐴 ran 𝑦)
91	90	ad4antr 732	. . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . 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 2940	. . . . . . . . . . . . . . . . . . 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 2988	. . . . . . . . . . . . . . . . . 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 5921	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑥 → ran 𝑦 = ran 𝑥)
96		rneq 5921	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑠 → ran 𝑦 = ran 𝑠)
97	95, 96	disji2 5108	. . . . . . . . . . . . . . . . . 18 ⊢ ((Disj 𝑦 ∈ 𝐴 ran 𝑦 ∧ (𝑥 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ 𝑥 ≠ 𝑠) → (ran 𝑥 ∩ ran 𝑠) = ∅)
98	91, 68, 48, 94, 97	syl121anc 1377	. . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . 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 5957	. . . . . . . . . . . . . . . 16 ⊢ (Rel 𝑠 → (𝑠 = ∅ ↔ ran 𝑠 = ∅))
102	101	biimpar 477	. . . . . . . . . . . . . . 15 ⊢ ((Rel 𝑠 ∧ ran 𝑠 = ∅) → 𝑠 = ∅)
103	59, 100, 102	syl2anc 584	. . . . . . . . . . . . . 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 14541	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ Word 𝐷 → 𝑥:(0..^(♯‘𝑥))⟶𝐷)
105		frel 6716	. . . . . . . . . . . . . . . 16 ⊢ (𝑥:(0..^(♯‘𝑥))⟶𝐷 → Rel 𝑥)
106	71, 104, 105	3syl 18	. . . . . . . . . . . . . . 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 5957	. . . . . . . . . . . . . . . 16 ⊢ (Rel 𝑥 → (𝑥 = ∅ ↔ ran 𝑥 = ∅))
108	107	biimpar 477	. . . . . . . . . . . . . . 15 ⊢ ((Rel 𝑥 ∧ ran 𝑥 = ∅) → 𝑥 = ∅)
109	106, 99, 108	syl2anc 584	. . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . 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 799	. . . . . . . . . . . 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 412	. . . . . . . . . . 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 466	. . . . . . . . . 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 3188	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑠 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))∀𝑥 ∈ (𝐴 ∖ (^◡♯ “ {0, 1}))(((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥) → 𝑠 = 𝑥))
115		dff13 7252	. . . . . . . . 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 583	. . . . . . . 8 ⊢ (𝜑 → (𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))):(𝐴 ∖ (^◡♯ “ {0, 1}))–1-1→(Base‘𝑆))
117		f1f1orn 6834	. . . . . . . 8 ⊢ ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))):(𝐴 ∖ (^◡♯ “ {0, 1}))–1-1→(Base‘𝑆) → (𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))):(𝐴 ∖ (^◡♯ “ {0, 1}))–1-1-onto→ran (𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))))
118	116, 117	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))):(𝐴 ∖ (^◡♯ “ {0, 1}))–1-1-onto→ran (𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))))
119		df-ima 5672	. . . . . . . . 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 6820	. . . . . . 7 ⊢ (𝜑 → ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))):(𝐴 ∖ (^◡♯ “ {0, 1}))–1-1-onto→(𝑀 “ (𝐴 ∖ (^◡♯ “ {0, 1}))) ↔ (𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))):(𝐴 ∖ (^◡♯ “ {0, 1}))–1-1-onto→ran (𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))))
122	118, 121	mpbird 257	. . . . . 6 ⊢ (𝜑 → (𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1}))):(𝐴 ∖ (^◡♯ “ {0, 1}))–1-1-onto→(𝑀 “ (𝐴 ∖ (^◡♯ “ {0, 1}))))
123		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) → 𝑐 = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥))
124	123	difeq1d 4105	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) → (𝑐 ∖ I ) = (((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥) ∖ I ))
125	124	dmeqd 5890	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 = ((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥)) → dom (𝑐 ∖ I ) = dom (((𝑀 ↾ (𝐴 ∖ (^◡♯ “ {0, 1})))‘𝑥) ∖ I ))
126	122, 125	disjrdx 32577	. . . . 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 232	. . . 4 ⊢ (𝜑 → Disj 𝑐 ∈ (𝑀 “ (𝐴 ∖ (^◡♯ “ {0, 1})))dom (𝑐 ∖ I ))
128		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∧ (𝑀‘𝑥) = 𝑐) → (𝑀‘𝑥) = 𝑐)
129	4	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∧ (𝑀‘𝑥) = 𝑐) → 𝐷 ∈ 𝑉)
130	14	ssrind 4224	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∩ (^◡♯ “ {0, 1})) ⊆ (dom 𝑀 ∩ (^◡♯ “ {0, 1})))
131	130	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∧ (𝑀‘𝑥) = 𝑐) → (𝐴 ∩ (^◡♯ “ {0, 1})) ⊆ (dom 𝑀 ∩ (^◡♯ “ {0, 1})))
132		simplr 768	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∧ (𝑀‘𝑥) = 𝑐) → 𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1})))
133	131, 132	sseldd 3964	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∧ (𝑀‘𝑥) = 𝑐) → 𝑥 ∈ (dom 𝑀 ∩ (^◡♯ “ {0, 1})))
134	5	tocyc01 33134	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑥 ∈ (dom 𝑀 ∩ (^◡♯ “ {0, 1}))) → (𝑀‘𝑥) = ( I ↾ 𝐷))
135	129, 133, 134	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∧ (𝑀‘𝑥) = 𝑐) → (𝑀‘𝑥) = ( I ↾ 𝐷))
136	128, 135	eqtr3d 2773	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∧ (𝑀‘𝑥) = 𝑐) → 𝑐 = ( I ↾ 𝐷))
137	136	difeq1d 4105	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∧ (𝑀‘𝑥) = 𝑐) → (𝑐 ∖ I ) = (( I ↾ 𝐷) ∖ I ))
138	137	dmeqd 5890	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∧ (𝑀‘𝑥) = 𝑐) → dom (𝑐 ∖ I ) = dom (( I ↾ 𝐷) ∖ I ))
139		resdifcom 5990	. . . . . . . . . 10 ⊢ (( I ↾ 𝐷) ∖ I ) = (( I ∖ I ) ↾ 𝐷)
140		difid 4356	. . . . . . . . . . 11 ⊢ ( I ∖ I ) = ∅
141	140	reseq1i 5967	. . . . . . . . . 10 ⊢ (( I ∖ I ) ↾ 𝐷) = (∅ ↾ 𝐷)
142		0res 32589	. . . . . . . . . 10 ⊢ (∅ ↾ 𝐷) = ∅
143	139, 141, 142	3eqtri 2763	. . . . . . . . 9 ⊢ (( I ↾ 𝐷) ∖ I ) = ∅
144	143	dmeqi 5889	. . . . . . . 8 ⊢ dom (( I ↾ 𝐷) ∖ I ) = dom ∅
145		dm0 5905	. . . . . . . 8 ⊢ dom ∅ = ∅
146	144, 145	eqtri 2759	. . . . . . 7 ⊢ dom (( I ↾ 𝐷) ∖ I ) = ∅
147	138, 146	eqtrdi 2787	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∧ (𝑀‘𝑥) = 𝑐) → dom (𝑐 ∖ I ) = ∅)
148	41	ffund 6715	. . . . . . 7 ⊢ (𝜑 → Fun 𝑀)
149		fvelima 6949	. . . . . . 7 ⊢ ((Fun 𝑀 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) → ∃𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))(𝑀‘𝑥) = 𝑐)
150	148, 149	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) → ∃𝑥 ∈ (𝐴 ∩ (^◡♯ “ {0, 1}))(𝑀‘𝑥) = 𝑐)
151	147, 150	r19.29a 3149	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) → dom (𝑐 ∖ I ) = ∅)
152	151	disjxun0 32560	. . . 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 257	. . 3 ⊢ (𝜑 → Disj 𝑐 ∈ ((𝑀 “ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1}))))dom (𝑐 ∖ I ))
154		uncom 4138	. . . . . 6 ⊢ ((𝑀 “ (𝐴 ∖ (^◡♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1})))) = ((𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∖ (^◡♯ “ {0, 1}))))
155		imaundi 6143	. . . . . 6 ⊢ (𝑀 “ ((𝐴 ∩ (^◡♯ “ {0, 1})) ∪ (𝐴 ∖ (^◡♯ “ {0, 1})))) = ((𝑀 “ (𝐴 ∩ (^◡♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∖ (^◡♯ “ {0, 1}))))
156		inundif 4459	. . . . . . 7 ⊢ ((𝐴 ∩ (^◡♯ “ {0, 1})) ∪ (𝐴 ∖ (^◡♯ “ {0, 1}))) = 𝐴
157	156	imaeq2i 6050	. . . . . 6 ⊢ (𝑀 “ ((𝐴 ∩ (^◡♯ “ {0, 1})) ∪ (𝐴 ∖ (^◡♯ “ {0, 1})))) = (𝑀 “ 𝐴)
158	154, 155, 157	3eqtr2i 2765	. . . . 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 5099	. . 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 232	. 2 ⊢ (𝜑 → Disj 𝑐 ∈ (𝑀 “ 𝐴)dom (𝑐 ∖ I ))
162	1, 2, 3, 8, 161	symgcntz 33101	1 ⊢ (𝜑 → (𝑀 “ 𝐴) ⊆ (𝑍‘(𝑀 “ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 ∀wral 3052 ∃wrex 3061 {crab 3420 Vcvv 3464 ∖ cdif 3928 ∪ cun 3929 ∩ cin 3930 ⊆ wss 3931 ∅c0 4313 {cpr 4608 Disj wdisj 5091 class class class wbr 5124 I cid 5552 ^◡ccnv 5658 dom cdm 5659 ran crn 5660 ↾ cres 5661 “ cima 5662 Rel wrel 5664 Fun wfun 6530 ⟶wf 6532 –1-1→wf1 6533 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7410 0cc0 11134 1c1 11135 < clt 11274 ..^cfzo 13676 ♯chash 14353 Word cword 14536 Basecbs 17233 Cntzccntz 19303 SymGrpcsymg 19355 toCycctocyc 33122

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-disj 5092 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9459 df-inf 9460 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-xnn0 12580 df-z 12594 df-uz 12858 df-rp 13014 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-hash 14354 df-word 14537 df-concat 14594 df-substr 14664 df-pfx 14694 df-csh 14812 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-tset 17295 df-efmnd 18852 df-cntz 19305 df-symg 19356 df-tocyc 33123

This theorem is referenced by: (None)

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