Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tocyccntz Structured version   Visualization version   GIF version

Theorem tocyccntz 31993
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 2736 . 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 31966 . . 3 (𝐷𝑉𝑀:{𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷}⟶(Base‘𝑆))
7 fimass 6689 . . 3 (𝑀:{𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷}⟶(Base‘𝑆) → (𝑀𝐴) ⊆ (Base‘𝑆))
84, 6, 73syl 18 . 2 (𝜑 → (𝑀𝐴) ⊆ (Base‘𝑆))
9 difss 4091 . . . . . . 7 (𝐴 ∖ (♯ “ {0, 1})) ⊆ 𝐴
10 tocyccntz.2 . . . . . . 7 (𝜑Disj 𝑥𝐴 ran 𝑥)
11 disjss1 5076 . . . . . . 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 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝐷𝑉)
14 tocyccntz.a . . . . . . . . . . . . . 14 (𝜑𝐴 ⊆ dom 𝑀)
1514adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝐴 ⊆ dom 𝑀)
16 simpr 485 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1})))
1716eldifad 3922 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝑥𝐴)
1815, 17sseldd 3945 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝑥 ∈ dom 𝑀)
19 fdm 6677 . . . . . . . . . . . . 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 2840 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝑥 ∈ {𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷})
22 id 22 . . . . . . . . . . . . 13 (𝑐 = 𝑥𝑐 = 𝑥)
23 dmeq 5859 . . . . . . . . . . . . 13 (𝑐 = 𝑥 → dom 𝑐 = dom 𝑥)
24 eqidd 2737 . . . . . . . . . . . . 13 (𝑐 = 𝑥𝐷 = 𝐷)
2522, 23, 24f1eq123d 6776 . . . . . . . . . . . 12 (𝑐 = 𝑥 → (𝑐:dom 𝑐1-1𝐷𝑥:dom 𝑥1-1𝐷))
2625elrab 3645 . . . . . . . . . . 11 (𝑥 ∈ {𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷} ↔ (𝑥 ∈ Word 𝐷𝑥:dom 𝑥1-1𝐷))
2721, 26sylib 217 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → (𝑥 ∈ Word 𝐷𝑥:dom 𝑥1-1𝐷))
2827simpld 495 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝑥 ∈ Word 𝐷)
2927simprd 496 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 𝑥:dom 𝑥1-1𝐷)
3016eldifbd 3923 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → ¬ 𝑥 ∈ (♯ “ {0, 1}))
31 hashgt1 31710 . . . . . . . . . . 11 (𝑥 ∈ V → (¬ 𝑥 ∈ (♯ “ {0, 1}) ↔ 1 < (♯‘𝑥)))
3231elv 3451 . . . . . . . . . 10 𝑥 ∈ (♯ “ {0, 1}) ↔ 1 < (♯‘𝑥))
3330, 32sylib 217 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → 1 < (♯‘𝑥))
345, 13, 28, 29, 33cycpmrn 31992 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → ran 𝑥 = dom ((𝑀𝑥) ∖ I ))
3516fvresd 6862 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) = (𝑀𝑥))
3635difeq1d 4081 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) ∖ I ) = ((𝑀𝑥) ∖ I ))
3736dmeqd 5861 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → dom (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) ∖ I ) = dom ((𝑀𝑥) ∖ I ))
3834, 37eqtr4d 2779 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → ran 𝑥 = dom (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) ∖ I ))
3938disjeq2dv 5075 . . . . . 6 (𝜑 → (Disj 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))ran 𝑥Disj 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))dom (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) ∖ I )))
4012, 39mpbid 231 . . . . 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 6699 . . . . . . . . . 10 (𝜑𝑀:dom 𝑀⟶(Base‘𝑆))
4314ssdifssd 4102 . . . . . . . . . 10 (𝜑 → (𝐴 ∖ (♯ “ {0, 1})) ⊆ dom 𝑀)
4442, 43fssresd 6709 . . . . . . . . 9 (𝜑 → (𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1}))):(𝐴 ∖ (♯ “ {0, 1}))⟶(Base‘𝑆))
4541, 14fssdmd 6687 . . . . . . . . . . . . . . . . . . . 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 3922 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠𝐴)
4946, 48sseldd 3945 . . . . . . . . . . . . . . . . . 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 5859 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = 𝑠 → dom 𝑐 = dom 𝑠)
52 eqidd 2737 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = 𝑠𝐷 = 𝐷)
5350, 51, 52f1eq123d 6776 . . . . . . . . . . . . . . . . . . 19 (𝑐 = 𝑠 → (𝑐:dom 𝑐1-1𝐷𝑠:dom 𝑠1-1𝐷))
5453elrab 3645 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ {𝑐 ∈ Word 𝐷𝑐:dom 𝑐1-1𝐷} ↔ (𝑠 ∈ Word 𝐷𝑠:dom 𝑠1-1𝐷))
5549, 54sylib 217 . . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . 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 14407 . . . . . . . . . . . . . . . 16 (𝑠 ∈ Word 𝐷𝑠:(0..^(♯‘𝑠))⟶𝐷)
58 frel 6673 . . . . . . . . . . . . . . . 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 6862 . . . . . . . . . . . . . . . . . . . 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 6862 . . . . . . . . . . . . . . . . . . . 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 2785 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → (𝑀𝑥) = (𝑀𝑠))
6564difeq1d 4081 . . . . . . . . . . . . . . . . . 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 5861 . . . . . . . . . . . . . . . . 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 3945 . . . . . . . . . . . . . . . . . . . 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 217 . . . . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . . . . 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 31992 . . . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . . . . 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 4100 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝐴 ∖ (♯ “ {0, 1})) ⊆ (dom 𝑀 ∖ (♯ “ {0, 1})))
7776sselda 3944 . . . . . . . . . . . . . . . . . . . . 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 3923 . . . . . . . . . . . . . . . . . . 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 31710 . . . . . . . . . . . . . . . . . . . 20 (𝑠𝐴 → (¬ 𝑠 ∈ (♯ “ {0, 1}) ↔ 1 < (♯‘𝑠)))
8180biimpa 477 . . . . . . . . . . . . . . . . . . 19 ((𝑠𝐴 ∧ ¬ 𝑠 ∈ (♯ “ {0, 1})) → 1 < (♯‘𝑠))
8248, 79, 81syl2anc 584 . . . . . . . . . . . . . . . . . 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 31992 . . . . . . . . . . . . . . . . 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 2787 . . . . . . . . . . . . . . . 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 4172 . . . . . . . . . . . . . . . . . 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 4178 . . . . . . . . . . . . . . . . . 18 (ran 𝑥 ∩ ran 𝑥) = ran 𝑥
8785, 86eqtrdi 2792 . . . . . . . . . . . . . . . . 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 5891 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑦 → ran 𝑥 = ran 𝑦)
8988cbvdisjv 5081 . . . . . . . . . . . . . . . . . . . 20 (Disj 𝑥𝐴 ran 𝑥Disj 𝑦𝐴 ran 𝑦)
9010, 89sylib 217 . . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → ¬ 𝑠 = 𝑥)
9392neqned 2950 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠𝑥)
9493necomd 2999 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑥𝑠)
95 rneq 5891 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑥 → ran 𝑦 = ran 𝑥)
96 rneq 5891 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑠 → ran 𝑦 = ran 𝑠)
9795, 96disji2 5087 . . . . . . . . . . . . . . . . . 18 ((Disj 𝑦𝐴 ran 𝑦 ∧ (𝑥𝐴𝑠𝐴) ∧ 𝑥𝑠) → (ran 𝑥 ∩ ran 𝑠) = ∅)
9891, 68, 48, 94, 97syl121anc 1375 . . . . . . . . . . . . . . . . 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 2778 . . . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . . . 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 5924 . . . . . . . . . . . . . . . 16 (Rel 𝑠 → (𝑠 = ∅ ↔ ran 𝑠 = ∅))
102101biimpar 478 . . . . . . . . . . . . . . 15 ((Rel 𝑠 ∧ ran 𝑠 = ∅) → 𝑠 = ∅)
10359, 100, 102syl2anc 584 . . . . . . . . . . . . . 14 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑠 = ∅)
104 wrdf 14407 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Word 𝐷𝑥:(0..^(♯‘𝑥))⟶𝐷)
105 frel 6673 . . . . . . . . . . . . . . . 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 5924 . . . . . . . . . . . . . . . 16 (Rel 𝑥 → (𝑥 = ∅ ↔ ran 𝑥 = ∅))
108107biimpar 478 . . . . . . . . . . . . . . 15 ((Rel 𝑥 ∧ ran 𝑥 = ∅) → 𝑥 = ∅)
109106, 99, 108syl2anc 584 . . . . . . . . . . . . . 14 (((((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) ∧ ¬ 𝑠 = 𝑥) → 𝑥 = ∅)
110103, 109eqtr4d 2779 . . . . . . . . . . . . 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 413 . . . . . . . . . . 11 (((𝜑𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))) → (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) → 𝑠 = 𝑥))
113112anasss 467 . . . . . . . . . 10 ((𝜑 ∧ (𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1})) ∧ 𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1})))) → (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) → 𝑠 = 𝑥))
114113ralrimivva 3197 . . . . . . . . 9 (𝜑 → ∀𝑠 ∈ (𝐴 ∖ (♯ “ {0, 1}))∀𝑥 ∈ (𝐴 ∖ (♯ “ {0, 1}))(((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑠) = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) → 𝑠 = 𝑥))
115 dff13 7202 . . . . . . . . 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 583 . . . . . . . 8 (𝜑 → (𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1}))):(𝐴 ∖ (♯ “ {0, 1}))–1-1→(Base‘𝑆))
117 f1f1orn 6795 . . . . . . . 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 5646 . . . . . . . . 9 (𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))) = ran (𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))
120119a1i 11 . . . . . . . 8 (𝜑 → (𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))) = ran (𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1}))))
121120f1oeq3d 6781 . . . . . . 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 256 . . . . . 6 (𝜑 → (𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1}))):(𝐴 ∖ (♯ “ {0, 1}))–1-1-onto→(𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))))
123 simpr 485 . . . . . . . 8 ((𝜑𝑐 = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) → 𝑐 = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥))
124123difeq1d 4081 . . . . . . 7 ((𝜑𝑐 = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) → (𝑐 ∖ I ) = (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) ∖ I ))
125124dmeqd 5861 . . . . . 6 ((𝜑𝑐 = ((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥)) → dom (𝑐 ∖ I ) = dom (((𝑀 ↾ (𝐴 ∖ (♯ “ {0, 1})))‘𝑥) ∖ I ))
126122, 125disjrdx 31509 . . . . 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 231 . . . 4 (𝜑Disj 𝑐 ∈ (𝑀 “ (𝐴 ∖ (♯ “ {0, 1})))dom (𝑐 ∖ I ))
128 simpr 485 . . . . . . . . . 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 4195 . . . . . . . . . . . . 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 3945 . . . . . . . . . . 11 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → 𝑥 ∈ (dom 𝑀 ∩ (♯ “ {0, 1})))
1345tocyc01 31967 . . . . . . . . . . 11 ((𝐷𝑉𝑥 ∈ (dom 𝑀 ∩ (♯ “ {0, 1}))) → (𝑀𝑥) = ( I ↾ 𝐷))
135129, 133, 134syl2anc 584 . . . . . . . . . 10 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → (𝑀𝑥) = ( I ↾ 𝐷))
136128, 135eqtr3d 2778 . . . . . . . . 9 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → 𝑐 = ( I ↾ 𝐷))
137136difeq1d 4081 . . . . . . . 8 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → (𝑐 ∖ I ) = (( I ↾ 𝐷) ∖ I ))
138137dmeqd 5861 . . . . . . 7 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → dom (𝑐 ∖ I ) = dom (( I ↾ 𝐷) ∖ I ))
139 resdifcom 5956 . . . . . . . . . 10 (( I ↾ 𝐷) ∖ I ) = (( I ∖ I ) ↾ 𝐷)
140 difid 4330 . . . . . . . . . . 11 ( I ∖ I ) = ∅
141140reseq1i 5933 . . . . . . . . . 10 (( I ∖ I ) ↾ 𝐷) = (∅ ↾ 𝐷)
142 0res 31522 . . . . . . . . . 10 (∅ ↾ 𝐷) = ∅
143139, 141, 1423eqtri 2768 . . . . . . . . 9 (( I ↾ 𝐷) ∖ I ) = ∅
144143dmeqi 5860 . . . . . . . 8 dom (( I ↾ 𝐷) ∖ I ) = dom ∅
145 dm0 5876 . . . . . . . 8 dom ∅ = ∅
146144, 145eqtri 2764 . . . . . . 7 dom (( I ↾ 𝐷) ∖ I ) = ∅
147138, 146eqtrdi 2792 . . . . . 6 ((((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) ∧ 𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))) ∧ (𝑀𝑥) = 𝑐) → dom (𝑐 ∖ I ) = ∅)
14841ffund 6672 . . . . . . 7 (𝜑 → Fun 𝑀)
149 fvelima 6908 . . . . . . 7 ((Fun 𝑀𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) → ∃𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))(𝑀𝑥) = 𝑐)
150148, 149sylan 580 . . . . . 6 ((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) → ∃𝑥 ∈ (𝐴 ∩ (♯ “ {0, 1}))(𝑀𝑥) = 𝑐)
151147, 150r19.29a 3159 . . . . 5 ((𝜑𝑐 ∈ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) → dom (𝑐 ∖ I ) = ∅)
152151disjxun0 31492 . . . 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 256 . . 3 (𝜑Disj 𝑐 ∈ ((𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1}))))dom (𝑐 ∖ I ))
154 uncom 4113 . . . . . 6 ((𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) = ((𝑀 “ (𝐴 ∩ (♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))))
155 imaundi 6102 . . . . . 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
157156imaeq2i 6011 . . . . . 6 (𝑀 “ ((𝐴 ∩ (♯ “ {0, 1})) ∪ (𝐴 ∖ (♯ “ {0, 1})))) = (𝑀𝐴)
158154, 155, 1573eqtr2i 2770 . . . . 5 ((𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) = (𝑀𝐴)
159158a1i 11 . . . 4 (𝜑 → ((𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1})))) = (𝑀𝐴))
160159disjeq1d 5078 . . 3 (𝜑 → (Disj 𝑐 ∈ ((𝑀 “ (𝐴 ∖ (♯ “ {0, 1}))) ∪ (𝑀 “ (𝐴 ∩ (♯ “ {0, 1}))))dom (𝑐 ∖ I ) ↔ Disj 𝑐 ∈ (𝑀𝐴)dom (𝑐 ∖ I )))
161153, 160mpbid 231 . 2 (𝜑Disj 𝑐 ∈ (𝑀𝐴)dom (𝑐 ∖ I ))
1621, 2, 3, 8, 161symgcntz 31936 1 (𝜑 → (𝑀𝐴) ⊆ (𝑍‘(𝑀𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  cdif 3907  cun 3908  cin 3909  wss 3910  c0 4282  {cpr 4588  Disj wdisj 5070   class class class wbr 5105   I cid 5530  ccnv 5632  dom cdm 5633  ran crn 5634  cres 5635  cima 5636  Rel wrel 5638  Fun wfun 6490  wf 6492  1-1wf1 6493  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  0cc0 11051  1c1 11052   < clt 11189  ..^cfzo 13567  chash 14230  Word cword 14402  Basecbs 17083  Cntzccntz 19095  SymGrpcsymg 19148  toCycctocyc 31955
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-hash 14231  df-word 14403  df-concat 14459  df-substr 14529  df-pfx 14559  df-csh 14677  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-tset 17152  df-efmnd 18679  df-cntz 19097  df-symg 19149  df-tocyc 31956
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator