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Theorem tocyccntz 30937
 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 (𝜑𝐷𝑉)
tocyccntz.2 (𝜑Disj 𝑥𝐴 ran 𝑥)
tocyccntz.a (𝜑𝐴 ⊆ dom 𝑀)
Assertion
Ref Expression
tocyccntz (𝜑 → (𝑀𝐴) ⊆ (𝑍‘(𝑀𝐴)))
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.
StepHypRef Expression
1 tocyccntz.s . 2 𝑆 = (SymGrp‘𝐷)
2 eqid 2758 . 2 (Base‘𝑆) = (Base‘𝑆)
3 tocyccntz.z . 2 𝑍 = (Cntz‘𝑆)
4 tocyccntz.1 . . 3 (𝜑𝐷𝑉)
5 tocyccntz.m . . . 4 𝑀 = (toCyc‘𝐷)
65, 1, 2tocycf 30910 . . 3 (𝐷𝑉𝑀:{𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷}⟶(Base‘𝑆))
7 fimass 6540 . . 3 (𝑀:{𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷}⟶(Base‘𝑆) → (𝑀𝐴) ⊆ (Base‘𝑆))
84, 6, 73syl 18 . 2 (𝜑 → (𝑀𝐴) ⊆ (Base‘𝑆))
9 difss 4037 . . . . . . 7 (𝐴 ∖ (♯ “ {0, 1})) ⊆ 𝐴
10 tocyccntz.2 . . . . . . 7 (𝜑Disj 𝑥𝐴 ran 𝑥)
11 disjss1 5003 . . . . . . 7 ((𝐴 ∖ (♯ “ {0, 1})) ⊆ 𝐴 → (Disj 𝑥𝐴 ran 𝑥Disj 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))ran 𝑥))
129, 10, 11mpsyl 68 . . . . . 6 (𝜑Disj 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))ran 𝑥)
134adantr 484 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝐷𝑉)
14 tocyccntz.a . . . . . . . . . . . . . 14 (𝜑𝐴 ⊆ dom 𝑀)
1514adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝐴 ⊆ dom 𝑀)
16 simpr 488 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1})))
1716eldifad 3870 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝑥𝐴)
1815, 17sseldd 3893 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝑥 ∈ dom 𝑀)
19 fdm 6506 . . . . . . . . . . . . 13 (𝑀:{𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷}⟶(Base‘𝑆) → dom 𝑀 = {𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷})
2013, 6, 193syl 18 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → dom 𝑀 = {𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷})
2118, 20eleqtrd 2854 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝑥 ∈ {𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷})
22 id 22 . . . . . . . . . . . . 13 (𝑐 = 𝑥𝑐 = 𝑥)
23 dmeq 5743 . . . . . . . . . . . . 13 (𝑐 = 𝑥 → dom 𝑐 = dom 𝑥)
24 eqidd 2759 . . . . . . . . . . . . 13 (𝑐 = 𝑥𝐷 = 𝐷)
2522, 23, 24f1eq123d 6594 . . . . . . . . . . . 12 (𝑐 = 𝑥 → (𝑐:dom 𝑐1-1𝐷𝑥:dom 𝑥1-1𝐷))
2625elrab 3602 . . . . . . . . . . 11 (𝑥 ∈ {𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷} ↔ (𝑥 ∈ Word 𝐷𝑥:dom 𝑥1-1𝐷))
2721, 26sylib 221 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → (𝑥 ∈ Word 𝐷𝑥:dom 𝑥1-1𝐷))
2827simpld 498 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝑥 ∈ Word 𝐷)
2927simprd 499 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝑥:dom 𝑥1-1𝐷)
3016eldifbd 3871 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → ¬ 𝑥 ∈ (♯ “ {0, 1}))
31 hashgt1 30652 . . . . . . . . . . 11 (𝑥 ∈ V → (¬ 𝑥 ∈ (♯ “ {0, 1}) ↔ 1 < (♯‘𝑥)))
3231elv 3415 . . . . . . . . . 10 𝑥 ∈ (♯ “ {0, 1}) ↔ 1 < (♯‘𝑥))
3330, 32sylib 221 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 1 < (♯‘𝑥))
345, 13, 28, 29, 33cycpmrn 30936 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → ran 𝑥 = dom ((𝑀𝑥) ∖ I ))
3516fvresd 6678 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) = (𝑀𝑥))
3635difeq1d 4027 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) ∖ I ) = ((𝑀𝑥) ∖ I ))
3736dmeqd 5745 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → dom (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) ∖ I ) = dom ((𝑀𝑥) ∖ I ))
3834, 37eqtr4d 2796 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → ran 𝑥 = dom (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) ∖ I ))
3938disjeq2dv 5002 . . . . . 6 (𝜑 → (Disj 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))ran 𝑥Disj 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))dom (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) ∖ I )))
4012, 39mpbid 235 . . . . 5 (𝜑Disj 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))dom (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) ∖ I ))
414, 6syl 17 . . . . . . . . . . 11 (𝜑𝑀:{𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷}⟶(Base‘𝑆))
4241ffdmd 6522 . . . . . . . . . 10 (𝜑𝑀:dom 𝑀⟶(Base‘𝑆))
4314ssdifssd 4048 . . . . . . . . . 10 (𝜑 → (𝐴 ∖ (♯ “ {0, 1})) ⊆ dom 𝑀)
4442, 43fssresd 6530 . . . . . . . . 9 (𝜑 → (𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1}))):(𝐴 ∖ (♯ “ {0, 1}))⟶(Base‘𝑆))
4541, 14fssdmd 6514 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐴 ⊆ {𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷})
4645ad4antr 731 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝐴 ⊆ {𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷})
47 simp-4r 783 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1})))
4847eldifad 3870 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠𝐴)
4946, 48sseldd 3893 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠 ∈ {𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷})
50 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = 𝑠𝑐 = 𝑠)
51 dmeq 5743 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = 𝑠 → dom 𝑐 = dom 𝑠)
52 eqidd 2759 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = 𝑠𝐷 = 𝐷)
5350, 51, 52f1eq123d 6594 . . . . . . . . . . . . . . . . . . 19 (𝑐 = 𝑠 → (𝑐:dom 𝑐1-1𝐷𝑠:dom 𝑠1-1𝐷))
5453elrab 3602 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ {𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷} ↔ (𝑠 ∈ Word 𝐷𝑠:dom 𝑠1-1𝐷))
5549, 54sylib 221 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → (𝑠 ∈ Word 𝐷𝑠:dom 𝑠1-1𝐷))
5655simpld 498 . . . . . . . . . . . . . . . 16 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠 ∈ Word 𝐷)
57 wrdf 13918 . . . . . . . . . . . . . . . 16 (𝑠 ∈ Word 𝐷𝑠:(0..^(♯‘𝑠))⟶𝐷)
58 frel 6503 . . . . . . . . . . . . . . . 16 (𝑠:(0..^(♯‘𝑠))⟶𝐷 → Rel 𝑠)
5956, 57, 583syl 18 . . . . . . . . . . . . . . 15 (((((𝜑𝑠 ∈ (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 (((((𝜑𝑠 ∈ (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))
6147fvresd 6678 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = (𝑀𝑠))
6216ad5ant13 756 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1})))
6362fvresd 6678 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) = (𝑀𝑥))
6460, 61, 633eqtr3rd 2802 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → (𝑀𝑥) = (𝑀𝑠))
6564difeq1d 4027 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ((𝑀𝑥) ∖ I ) = ((𝑀𝑠) ∖ I ))
6665dmeqd 5745 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → dom ((𝑀𝑥) ∖ I ) = dom ((𝑀𝑠) ∖ I ))
674ad4antr 731 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝐷𝑉)
6817ad5ant13 756 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑥𝐴)
6946, 68sseldd 3893 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑥 ∈ {𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷})
7069, 26sylib 221 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → (𝑥 ∈ Word 𝐷𝑥:dom 𝑥1-1𝐷))
7170simpld 498 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑥 ∈ Word 𝐷)
7270simprd 499 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑥:dom 𝑥1-1𝐷)
7333ad5ant13 756 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 1 < (♯‘𝑥))
745, 67, 71, 72, 73cycpmrn 30936 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ran 𝑥 = dom ((𝑀𝑥) ∖ I ))
7555simprd 499 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠:dom 𝑠1-1𝐷)
7614ssdifd 4046 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝐴 ∖ (♯ “ {0, 1})) ⊆ (dom 𝑀 ∖ (♯ “ {0, 1})))
7776sselda 3892 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝑠 ∈ (dom 𝑀 ∖ (♯ “ {0, 1})))
7877ad3antrrr 729 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠 ∈ (dom 𝑀 ∖ (♯ “ {0, 1})))
7978eldifbd 3871 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ¬ 𝑠 ∈ (♯ “ {0, 1}))
80 hashgt1 30652 . . . . . . . . . . . . . . . . . . . 20 (𝑠𝐴 → (¬ 𝑠 ∈ (♯ “ {0, 1}) ↔ 1 < (♯‘𝑠)))
8180biimpa 480 . . . . . . . . . . . . . . . . . . 19 ((𝑠𝐴 ∧ ¬ 𝑠 ∈ (♯ “ {0, 1})) → 1 < (♯‘𝑠))
8248, 79, 81syl2anc 587 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 1 < (♯‘𝑠))
835, 67, 56, 75, 82cycpmrn 30936 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ran 𝑠 = dom ((𝑀𝑠) ∖ I ))
8466, 74, 833eqtr4rd 2804 . . . . . . . . . . . . . . . 16 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ran 𝑠 = ran 𝑥)
8584ineq2d 4117 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑠 ∈ (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 4123 . . . . . . . . . . . . . . . . . 18 (ran 𝑥 ∩ ran 𝑥) = ran 𝑥
8785, 86eqtrdi 2809 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑠 ∈ (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 5777 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑦 → ran 𝑥 = ran 𝑦)
8988cbvdisjv 5008 . . . . . . . . . . . . . . . . . . . 20 (Disj 𝑥𝐴 ran 𝑥Disj 𝑦𝐴 ran 𝑦)
9010, 89sylib 221 . . . . . . . . . . . . . . . . . . 19 (𝜑Disj 𝑦𝐴 ran 𝑦)
9190ad4antr 731 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → Disj 𝑦𝐴 ran 𝑦)
92 simpr 488 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ¬ 𝑠 = 𝑥)
9392neqned 2958 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠𝑥)
9493necomd 3006 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑥𝑠)
95 rneq 5777 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑥 → ran 𝑦 = ran 𝑥)
96 rneq 5777 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑠 → ran 𝑦 = ran 𝑠)
9795, 96disji2 5014 . . . . . . . . . . . . . . . . . 18 ((Disj 𝑦𝐴 ran 𝑦 ∧ (𝑥𝐴𝑠𝐴) ∧ 𝑥𝑠) → (ran 𝑥 ∩ ran 𝑠) = ∅)
9891, 68, 48, 94, 97syl121anc 1372 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → (ran 𝑥 ∩ ran 𝑠) = ∅)
9987, 98eqtr3d 2795 . . . . . . . . . . . . . . . 16 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ran 𝑥 = ∅)
10084, 99eqtrd 2793 . . . . . . . . . . . . . . 15 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ran 𝑠 = ∅)
101 relrn0 5810 . . . . . . . . . . . . . . . 16 (Rel 𝑠 → (𝑠 = ∅ ↔ ran 𝑠 = ∅))
102101biimpar 481 . . . . . . . . . . . . . . 15 ((Rel 𝑠 ∧ ran 𝑠 = ∅) → 𝑠 = ∅)
10359, 100, 102syl2anc 587 . . . . . . . . . . . . . 14 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠 = ∅)
104 wrdf 13918 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Word 𝐷𝑥:(0..^(♯‘𝑥))⟶𝐷)
105 frel 6503 . . . . . . . . . . . . . . . 16 (𝑥:(0..^(♯‘𝑥))⟶𝐷 → Rel 𝑥)
10671, 104, 1053syl 18 . . . . . . . . . . . . . . 15 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → Rel 𝑥)
107 relrn0 5810 . . . . . . . . . . . . . . . 16 (Rel 𝑥 → (𝑥 = ∅ ↔ ran 𝑥 = ∅))
108107biimpar 481 . . . . . . . . . . . . . . 15 ((Rel 𝑥 ∧ ran 𝑥 = ∅) → 𝑥 = ∅)
109106, 99, 108syl2anc 587 . . . . . . . . . . . . . 14 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑥 = ∅)
110103, 109eqtr4d 2796 . . . . . . . . . . . . 13 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠 = 𝑥)
111110pm2.18da 799 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) → 𝑠 = 𝑥)
112111ex 416 . . . . . . . . . . 11 (((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) → 𝑠 = 𝑥))
113112anasss 470 . . . . . . . . . 10 ((𝜑 ∧ (𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1})) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1})))) → (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) → 𝑠 = 𝑥))
114113ralrimivva 3120 . . . . . . . . 9 (𝜑 → ∀𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))∀𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))(((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) → 𝑠 = 𝑥))
115 dff13 7005 . . . . . . . . 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)))
11644, 114, 115sylanbrc 586 . . . . . . . 8 (𝜑 → (𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1}))):(𝐴 ∖ (♯ “ {0, 1}))–1-1→(Base‘𝑆))
117 f1f1orn 6613 . . . . . . . 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}))))
118116, 117syl 17 . . . . . . 7 (𝜑 → (𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1}))):(𝐴 ∖ (♯ “ {0, 1}))–1-1-onto→ran (𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1}))))
119 df-ima 5537 . . . . . . . . 9 (𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))) = ran (𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))
120119a1i 11 . . . . . . . 8 (𝜑 → (𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))) = ran (𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1}))))
121120f1oeq3d 6599 . . . . . . 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})))))
122118, 121mpbird 260 . . . . . 6 (𝜑 → (𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1}))):(𝐴 ∖ (♯ “ {0, 1}))–1-1-onto→(𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))))
123 simpr 488 . . . . . . . 8 ((𝜑𝑐 = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) → 𝑐 = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥))
124123difeq1d 4027 . . . . . . 7 ((𝜑𝑐 = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) → (𝑐 ∖ I ) = (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) ∖ I ))
125124dmeqd 5745 . . . . . 6 ((𝜑𝑐 = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) → dom (𝑐 ∖ I ) = dom (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) ∖ I ))
126122, 125disjrdx 30452 . . . . 5 (𝜑 → (Disj 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))dom (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) ∖ I ) ↔ Disj 𝑐 ∈ (𝑀 “ (𝐴 ∖ (♯ “ {0, 1})))dom (𝑐 ∖ I )))
12740, 126mpbid 235 . . . 4 (𝜑Disj 𝑐 ∈ (𝑀 “ (𝐴 ∖ (♯ “ {0, 1})))dom (𝑐 ∖ I ))
128 simpr 488 . . . . . . . . . 10 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → (𝑀𝑥) = 𝑐)
1294ad3antrrr 729 . . . . . . . . . . 11 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → 𝐷𝑉)
13014ssrind 4140 . . . . . . . . . . . . 13 (𝜑 → (𝐴 ∩ (♯ “ {0, 1})) ⊆ (dom 𝑀 ∩ (♯ “ {0, 1})))
131130ad3antrrr 729 . . . . . . . . . . . 12 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → (𝐴 ∩ (♯ “ {0, 1})) ⊆ (dom 𝑀 ∩ (♯ “ {0, 1})))
132 simplr 768 . . . . . . . . . . . 12 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1})))
133131, 132sseldd 3893 . . . . . . . . . . 11 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → 𝑥 ∈ (dom 𝑀 ∩ (♯ “ {0, 1})))
1345tocyc01 30911 . . . . . . . . . . 11 ((𝐷𝑉𝑥 ∈ (dom 𝑀 ∩ (♯ “ {0, 1}))) → (𝑀𝑥) = ( I ↾ 𝐷))
135129, 133, 134syl2anc 587 . . . . . . . . . 10 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → (𝑀𝑥) = ( I ↾ 𝐷))
136128, 135eqtr3d 2795 . . . . . . . . 9 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → 𝑐 = ( I ↾ 𝐷))
137136difeq1d 4027 . . . . . . . 8 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → (𝑐 ∖ I ) = (( I ↾ 𝐷) ∖ I ))
138137dmeqd 5745 . . . . . . 7 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → dom (𝑐 ∖ I ) = dom (( I ↾ 𝐷) ∖ I ))
139 resdifcom 5842 . . . . . . . . . 10 (( I ↾ 𝐷) ∖ I ) = (( I ∖ I ) ↾ 𝐷)
140 difid 4269 . . . . . . . . . . 11 ( I ∖ I ) = ∅
141140reseq1i 5819 . . . . . . . . . 10 (( I ∖ I ) ↾ 𝐷) = (∅ ↾ 𝐷)
142 0res 30465 . . . . . . . . . 10 (∅ ↾ 𝐷) = ∅
143139, 141, 1423eqtri 2785 . . . . . . . . 9 (( I ↾ 𝐷) ∖ I ) = ∅
144143dmeqi 5744 . . . . . . . 8 dom (( I ↾ 𝐷) ∖ I ) = dom ∅
145 dm0 5761 . . . . . . . 8 dom ∅ = ∅
146144, 145eqtri 2781 . . . . . . 7 dom (( I ↾ 𝐷) ∖ I ) = ∅
147138, 146eqtrdi 2809 . . . . . 6 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → dom (𝑐 ∖ I ) = ∅)
14841ffund 6502 . . . . . . 7 (𝜑 → Fun 𝑀)
149 fvelima 6719 . . . . . . 7 ((Fun 𝑀𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) → ∃𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))(𝑀𝑥) = 𝑐)
150148, 149sylan 583 . . . . . 6 ((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) → ∃𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))(𝑀𝑥) = 𝑐)
151147, 150r19.29a 3213 . . . . 5 ((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) → dom (𝑐 ∖ I ) = ∅)
152151disjxun0 30435 . . . 4 (𝜑 → (Disj 𝑐 ∈ ((𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1}))))dom (𝑐 ∖ I ) ↔ Disj 𝑐 ∈ (𝑀 “ (𝐴 ∖ (♯ “ {0, 1})))dom (𝑐 ∖ I )))
153127, 152mpbird 260 . . 3 (𝜑Disj 𝑐 ∈ ((𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1}))))dom (𝑐 ∖ I ))
154 uncom 4058 . . . . . 6 ((𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) = ((𝑀 “ (𝐴 ∩ (♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))))
155 imaundi 5980 . . . . . 6 (𝑀 “ ((𝐴 ∩ (♯ “ {0, 1})) ∪ (𝐴 ∖ (♯ “ {0, 1})))) = ((𝑀 “ (𝐴 ∩ (♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))))
156 inundif 4375 . . . . . . 7 ((𝐴 ∩ (♯ “ {0, 1})) ∪ (𝐴 ∖ (♯ “ {0, 1}))) = 𝐴
157156imaeq2i 5899 . . . . . 6 (𝑀 “ ((𝐴 ∩ (♯ “ {0, 1})) ∪ (𝐴 ∖ (♯ “ {0, 1})))) = (𝑀𝐴)
158154, 155, 1573eqtr2i 2787 . . . . 5 ((𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) = (𝑀𝐴)
159158a1i 11 . . . 4 (𝜑 → ((𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) = (𝑀𝐴))
160159disjeq1d 5005 . . 3 (𝜑 → (Disj 𝑐 ∈ ((𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1}))))dom (𝑐 ∖ I ) ↔ Disj 𝑐 ∈ (𝑀𝐴)dom (𝑐 ∖ I )))
161153, 160mpbid 235 . 2 (𝜑Disj 𝑐 ∈ (𝑀𝐴)dom (𝑐 ∖ I ))
1621, 2, 3, 8, 161symgcntz 30880 1 (𝜑 → (𝑀𝐴) ⊆ (𝑍‘(𝑀𝐴)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ∃wrex 3071  {crab 3074  Vcvv 3409   ∖ cdif 3855   ∪ cun 3856   ∩ cin 3857   ⊆ wss 3858  ∅c0 4225  {cpr 4524  Disj wdisj 4997   class class class wbr 5032   I cid 5429  ◡ccnv 5523  dom cdm 5524  ran crn 5525   ↾ cres 5526   “ cima 5527  Rel wrel 5529  Fun wfun 6329  ⟶wf 6331  –1-1→wf1 6332  –1-1-onto→wf1o 6334  ‘cfv 6335  (class class class)co 7150  0cc0 10575  1c1 10576   < clt 10713  ..^cfzo 13082  ♯chash 13740  Word cword 13913  Basecbs 16541  Cntzccntz 18512  SymGrpcsymg 18562  toCycctocyc 30899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652  ax-pre-sup 10653 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-disj 4998  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-er 8299  df-map 8418  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-sup 8939  df-inf 8940  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-div 11336  df-nn 11675  df-2 11737  df-3 11738  df-4 11739  df-5 11740  df-6 11741  df-7 11742  df-8 11743  df-9 11744  df-n0 11935  df-xnn0 12007  df-z 12021  df-uz 12283  df-rp 12431  df-fz 12940  df-fzo 13083  df-fl 13211  df-mod 13287  df-hash 13741  df-word 13914  df-concat 13970  df-substr 14050  df-pfx 14080  df-csh 14198  df-struct 16543  df-ndx 16544  df-slot 16545  df-base 16547  df-sets 16548  df-ress 16549  df-plusg 16636  df-tset 16642  df-efmnd 18100  df-cntz 18514  df-symg 18563  df-tocyc 30900 This theorem is referenced by: (None)
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