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Theorem tocyccntz 30781
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 2821 . 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 30754 . . 3 (𝐷𝑉𝑀:{𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷}⟶(Base‘𝑆))
7 fimass 6549 . . 3 (𝑀:{𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷}⟶(Base‘𝑆) → (𝑀𝐴) ⊆ (Base‘𝑆))
84, 6, 73syl 18 . 2 (𝜑 → (𝑀𝐴) ⊆ (Base‘𝑆))
9 difss 4107 . . . . . . 7 (𝐴 ∖ (♯ “ {0, 1})) ⊆ 𝐴
10 tocyccntz.2 . . . . . . 7 (𝜑Disj 𝑥𝐴 ran 𝑥)
11 disjss1 5029 . . . . . . 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 483 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝐷𝑉)
14 tocyccntz.a . . . . . . . . . . . . . 14 (𝜑𝐴 ⊆ dom 𝑀)
1514adantr 483 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝐴 ⊆ dom 𝑀)
16 simpr 487 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1})))
1716eldifad 3947 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝑥𝐴)
1815, 17sseldd 3967 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝑥 ∈ dom 𝑀)
19 fdm 6516 . . . . . . . . . . . . 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 2915 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝑥 ∈ {𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷})
22 id 22 . . . . . . . . . . . . 13 (𝑐 = 𝑥𝑐 = 𝑥)
23 dmeq 5766 . . . . . . . . . . . . 13 (𝑐 = 𝑥 → dom 𝑐 = dom 𝑥)
24 eqidd 2822 . . . . . . . . . . . . 13 (𝑐 = 𝑥𝐷 = 𝐷)
2522, 23, 24f1eq123d 6602 . . . . . . . . . . . 12 (𝑐 = 𝑥 → (𝑐:dom 𝑐1-1𝐷𝑥:dom 𝑥1-1𝐷))
2625elrab 3679 . . . . . . . . . . 11 (𝑥 ∈ {𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷} ↔ (𝑥 ∈ Word 𝐷𝑥:dom 𝑥1-1𝐷))
2721, 26sylib 220 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → (𝑥 ∈ Word 𝐷𝑥:dom 𝑥1-1𝐷))
2827simpld 497 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝑥 ∈ Word 𝐷)
2927simprd 498 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝑥:dom 𝑥1-1𝐷)
3016eldifbd 3948 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → ¬ 𝑥 ∈ (♯ “ {0, 1}))
31 hashgt1 30524 . . . . . . . . . . 11 (𝑥 ∈ V → (¬ 𝑥 ∈ (♯ “ {0, 1}) ↔ 1 < (♯‘𝑥)))
3231elv 3499 . . . . . . . . . 10 𝑥 ∈ (♯ “ {0, 1}) ↔ 1 < (♯‘𝑥))
3330, 32sylib 220 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 1 < (♯‘𝑥))
345, 13, 28, 29, 33cycpmrn 30780 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → ran 𝑥 = dom ((𝑀𝑥) ∖ I ))
3516fvresd 6684 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) = (𝑀𝑥))
3635difeq1d 4097 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) ∖ I ) = ((𝑀𝑥) ∖ I ))
3736dmeqd 5768 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → dom (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) ∖ I ) = dom ((𝑀𝑥) ∖ I ))
3834, 37eqtr4d 2859 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → ran 𝑥 = dom (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) ∖ I ))
3938disjeq2dv 5028 . . . . . 6 (𝜑 → (Disj 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))ran 𝑥Disj 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))dom (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) ∖ I )))
4012, 39mpbid 234 . . . . 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 6531 . . . . . . . . . 10 (𝜑𝑀:dom 𝑀⟶(Base‘𝑆))
4314ssdifssd 4118 . . . . . . . . . 10 (𝜑 → (𝐴 ∖ (♯ “ {0, 1})) ⊆ dom 𝑀)
4442, 43fssresd 6539 . . . . . . . . 9 (𝜑 → (𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1}))):(𝐴 ∖ (♯ “ {0, 1}))⟶(Base‘𝑆))
4541, 14fssdmd 6523 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐴 ⊆ {𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷})
4645ad4antr 730 . . . . . . . . . . . . . . . . . . 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 782 . . . . . . . . . . . . . . . . . . . 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 3947 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠𝐴)
4946, 48sseldd 3967 . . . . . . . . . . . . . . . . . 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 5766 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = 𝑠 → dom 𝑐 = dom 𝑠)
52 eqidd 2822 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = 𝑠𝐷 = 𝐷)
5350, 51, 52f1eq123d 6602 . . . . . . . . . . . . . . . . . . 19 (𝑐 = 𝑠 → (𝑐:dom 𝑐1-1𝐷𝑠:dom 𝑠1-1𝐷))
5453elrab 3679 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ {𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷} ↔ (𝑠 ∈ Word 𝐷𝑠:dom 𝑠1-1𝐷))
5549, 54sylib 220 . . . . . . . . . . . . . . . . 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 497 . . . . . . . . . . . . . . . 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 13860 . . . . . . . . . . . . . . . 16 (𝑠 ∈ Word 𝐷𝑠:(0..^(♯‘𝑠))⟶𝐷)
58 frel 6513 . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . . . . 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 6684 . . . . . . . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . . . . . . . . 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 6684 . . . . . . . . . . . . . . . . . . . 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 2865 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → (𝑀𝑥) = (𝑀𝑠))
6564difeq1d 4097 . . . . . . . . . . . . . . . . . 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 5768 . . . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝐷𝑉)
6817ad5ant13 755 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑥𝐴)
6946, 68sseldd 3967 . . . . . . . . . . . . . . . . . . . 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 220 . . . . . . . . . . . . . . . . . . 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 497 . . . . . . . . . . . . . . . . . 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 498 . . . . . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . . . . . 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 30780 . . . . . . . . . . . . . . . . 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 498 . . . . . . . . . . . . . . . . . 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 4116 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝐴 ∖ (♯ “ {0, 1})) ⊆ (dom 𝑀 ∖ (♯ “ {0, 1})))
7776sselda 3966 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝑠 ∈ (dom 𝑀 ∖ (♯ “ {0, 1})))
7877ad3antrrr 728 . . . . . . . . . . . . . . . . . . . 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 3948 . . . . . . . . . . . . . . . . . . 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 30524 . . . . . . . . . . . . . . . . . . . 20 (𝑠𝐴 → (¬ 𝑠 ∈ (♯ “ {0, 1}) ↔ 1 < (♯‘𝑠)))
8180biimpa 479 . . . . . . . . . . . . . . . . . . 19 ((𝑠𝐴 ∧ ¬ 𝑠 ∈ (♯ “ {0, 1})) → 1 < (♯‘𝑠))
8248, 79, 81syl2anc 586 . . . . . . . . . . . . . . . . . 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 30780 . . . . . . . . . . . . . . . . 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 2867 . . . . . . . . . . . . . . . 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 4188 . . . . . . . . . . . . . . . . . 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 4194 . . . . . . . . . . . . . . . . . 18 (ran 𝑥 ∩ ran 𝑥) = ran 𝑥
8785, 86syl6eq 2872 . . . . . . . . . . . . . . . . 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 5800 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑦 → ran 𝑥 = ran 𝑦)
8988cbvdisjv 5034 . . . . . . . . . . . . . . . . . . . 20 (Disj 𝑥𝐴 ran 𝑥Disj 𝑦𝐴 ran 𝑦)
9010, 89sylib 220 . . . . . . . . . . . . . . . . . . 19 (𝜑Disj 𝑦𝐴 ran 𝑦)
9190ad4antr 730 . . . . . . . . . . . . . . . . . 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 487 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ¬ 𝑠 = 𝑥)
9392neqned 3023 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠𝑥)
9493necomd 3071 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑥𝑠)
95 rneq 5800 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑥 → ran 𝑦 = ran 𝑥)
96 rneq 5800 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑠 → ran 𝑦 = ran 𝑠)
9795, 96disji2 5040 . . . . . . . . . . . . . . . . . 18 ((Disj 𝑦𝐴 ran 𝑦 ∧ (𝑥𝐴𝑠𝐴) ∧ 𝑥𝑠) → (ran 𝑥 ∩ ran 𝑠) = ∅)
9891, 68, 48, 94, 97syl121anc 1371 . . . . . . . . . . . . . . . . 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 2858 . . . . . . . . . . . . . . . 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 2856 . . . . . . . . . . . . . . 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 5834 . . . . . . . . . . . . . . . 16 (Rel 𝑠 → (𝑠 = ∅ ↔ ran 𝑠 = ∅))
102101biimpar 480 . . . . . . . . . . . . . . 15 ((Rel 𝑠 ∧ ran 𝑠 = ∅) → 𝑠 = ∅)
10359, 100, 102syl2anc 586 . . . . . . . . . . . . . 14 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠 = ∅)
104 wrdf 13860 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Word 𝐷𝑥:(0..^(♯‘𝑥))⟶𝐷)
105 frel 6513 . . . . . . . . . . . . . . . 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 5834 . . . . . . . . . . . . . . . 16 (Rel 𝑥 → (𝑥 = ∅ ↔ ran 𝑥 = ∅))
108107biimpar 480 . . . . . . . . . . . . . . 15 ((Rel 𝑥 ∧ ran 𝑥 = ∅) → 𝑥 = ∅)
109106, 99, 108syl2anc 586 . . . . . . . . . . . . . 14 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑥 = ∅)
110103, 109eqtr4d 2859 . . . . . . . . . . . . 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 798 . . . . . . . . . . . 12 ((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) → 𝑠 = 𝑥)
112111ex 415 . . . . . . . . . . 11 (((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) → 𝑠 = 𝑥))
113112anasss 469 . . . . . . . . . 10 ((𝜑 ∧ (𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1})) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1})))) → (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) → 𝑠 = 𝑥))
114113ralrimivva 3191 . . . . . . . . 9 (𝜑 → ∀𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))∀𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))(((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) → 𝑠 = 𝑥))
115 dff13 7007 . . . . . . . . 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 585 . . . . . . . 8 (𝜑 → (𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1}))):(𝐴 ∖ (♯ “ {0, 1}))–1-1→(Base‘𝑆))
117 f1f1orn 6620 . . . . . . . 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 5562 . . . . . . . . 9 (𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))) = ran (𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))
120119a1i 11 . . . . . . . 8 (𝜑 → (𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))) = ran (𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1}))))
121120f1oeq3d 6606 . . . . . . 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 259 . . . . . 6 (𝜑 → (𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1}))):(𝐴 ∖ (♯ “ {0, 1}))–1-1-onto→(𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))))
123 simpr 487 . . . . . . . 8 ((𝜑𝑐 = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) → 𝑐 = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥))
124123difeq1d 4097 . . . . . . 7 ((𝜑𝑐 = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) → (𝑐 ∖ I ) = (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) ∖ I ))
125124dmeqd 5768 . . . . . 6 ((𝜑𝑐 = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) → dom (𝑐 ∖ I ) = dom (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) ∖ I ))
126122, 125disjrdx 30335 . . . . 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 234 . . . 4 (𝜑Disj 𝑐 ∈ (𝑀 “ (𝐴 ∖ (♯ “ {0, 1})))dom (𝑐 ∖ I ))
128 simpr 487 . . . . . . . . . 10 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → (𝑀𝑥) = 𝑐)
1294ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → 𝐷𝑉)
13014ssrind 4211 . . . . . . . . . . . . 13 (𝜑 → (𝐴 ∩ (♯ “ {0, 1})) ⊆ (dom 𝑀 ∩ (♯ “ {0, 1})))
131130ad3antrrr 728 . . . . . . . . . . . 12 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → (𝐴 ∩ (♯ “ {0, 1})) ⊆ (dom 𝑀 ∩ (♯ “ {0, 1})))
132 simplr 767 . . . . . . . . . . . 12 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1})))
133131, 132sseldd 3967 . . . . . . . . . . 11 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → 𝑥 ∈ (dom 𝑀 ∩ (♯ “ {0, 1})))
1345tocyc01 30755 . . . . . . . . . . 11 ((𝐷𝑉𝑥 ∈ (dom 𝑀 ∩ (♯ “ {0, 1}))) → (𝑀𝑥) = ( I ↾ 𝐷))
135129, 133, 134syl2anc 586 . . . . . . . . . 10 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → (𝑀𝑥) = ( I ↾ 𝐷))
136128, 135eqtr3d 2858 . . . . . . . . 9 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → 𝑐 = ( I ↾ 𝐷))
137136difeq1d 4097 . . . . . . . 8 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → (𝑐 ∖ I ) = (( I ↾ 𝐷) ∖ I ))
138137dmeqd 5768 . . . . . . 7 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → dom (𝑐 ∖ I ) = dom (( I ↾ 𝐷) ∖ I ))
139 resdifcom 5866 . . . . . . . . . 10 (( I ↾ 𝐷) ∖ I ) = (( I ∖ I ) ↾ 𝐷)
140 difid 4329 . . . . . . . . . . 11 ( I ∖ I ) = ∅
141140reseq1i 5843 . . . . . . . . . 10 (( I ∖ I ) ↾ 𝐷) = (∅ ↾ 𝐷)
142 0res 30348 . . . . . . . . . 10 (∅ ↾ 𝐷) = ∅
143139, 141, 1423eqtri 2848 . . . . . . . . 9 (( I ↾ 𝐷) ∖ I ) = ∅
144143dmeqi 5767 . . . . . . . 8 dom (( I ↾ 𝐷) ∖ I ) = dom ∅
145 dm0 5784 . . . . . . . 8 dom ∅ = ∅
146144, 145eqtri 2844 . . . . . . 7 dom (( I ↾ 𝐷) ∖ I ) = ∅
147138, 146syl6eq 2872 . . . . . 6 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → dom (𝑐 ∖ I ) = ∅)
14841ffund 6512 . . . . . . 7 (𝜑 → Fun 𝑀)
149 fvelima 6725 . . . . . . 7 ((Fun 𝑀𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) → ∃𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))(𝑀𝑥) = 𝑐)
150148, 149sylan 582 . . . . . 6 ((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) → ∃𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))(𝑀𝑥) = 𝑐)
151147, 150r19.29a 3289 . . . . 5 ((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) → dom (𝑐 ∖ I ) = ∅)
152151disjxun0 30318 . . . 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 259 . . 3 (𝜑Disj 𝑐 ∈ ((𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1}))))dom (𝑐 ∖ I ))
154 uncom 4128 . . . . . 6 ((𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) = ((𝑀 “ (𝐴 ∩ (♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))))
155 imaundi 6002 . . . . . 6 (𝑀 “ ((𝐴 ∩ (♯ “ {0, 1})) ∪ (𝐴 ∖ (♯ “ {0, 1})))) = ((𝑀 “ (𝐴 ∩ (♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))))
156 inundif 4426 . . . . . . 7 ((𝐴 ∩ (♯ “ {0, 1})) ∪ (𝐴 ∖ (♯ “ {0, 1}))) = 𝐴
157156imaeq2i 5921 . . . . . 6 (𝑀 “ ((𝐴 ∩ (♯ “ {0, 1})) ∪ (𝐴 ∖ (♯ “ {0, 1})))) = (𝑀𝐴)
158154, 155, 1573eqtr2i 2850 . . . . 5 ((𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) = (𝑀𝐴)
159158a1i 11 . . . 4 (𝜑 → ((𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) = (𝑀𝐴))
160159disjeq1d 5031 . . 3 (𝜑 → (Disj 𝑐 ∈ ((𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1}))))dom (𝑐 ∖ I ) ↔ Disj 𝑐 ∈ (𝑀𝐴)dom (𝑐 ∖ I )))
161153, 160mpbid 234 . 2 (𝜑Disj 𝑐 ∈ (𝑀𝐴)dom (𝑐 ∖ I ))
1621, 2, 3, 8, 161symgcntz 30724 1 (𝜑 → (𝑀𝐴) ⊆ (𝑍‘(𝑀𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  wne 3016  wral 3138  wrex 3139  {crab 3142  Vcvv 3494  cdif 3932  cun 3933  cin 3934  wss 3935  c0 4290  {cpr 4562  Disj wdisj 5023   class class class wbr 5058   I cid 5453  ccnv 5548  dom cdm 5549  ran crn 5550  cres 5551  cima 5552  Rel wrel 5554  Fun wfun 6343  wf 6345  1-1wf1 6346  1-1-ontowf1o 6348  cfv 6349  (class class class)co 7150  0cc0 10531  1c1 10532   < clt 10669  ..^cfzo 13027  chash 13684  Word cword 13855  Basecbs 16477  Cntzccntz 18439  SymGrpcsymg 18489  toCycctocyc 30743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-disj 5024  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-sup 8900  df-inf 8901  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-xnn0 11962  df-z 11976  df-uz 12238  df-rp 12384  df-fz 12887  df-fzo 13028  df-fl 13156  df-mod 13232  df-hash 13685  df-word 13856  df-concat 13917  df-substr 13997  df-pfx 14027  df-csh 14145  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-plusg 16572  df-tset 16578  df-efmnd 18028  df-cntz 18441  df-symg 18490  df-tocyc 30744
This theorem is referenced by: (None)
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