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Theorem zfz1iso 10821
Description: A finite set of integers has an order isomorphism to a one-based finite sequence. (Contributed by Jim Kingdon, 3-Sep-2022.)
Assertion
Ref Expression
zfz1iso ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴))
Distinct variable group:   𝐴,𝑓

Proof of Theorem zfz1iso
Dummy variables 𝑛 𝑥 𝑎 𝑘 𝑚 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isfi 6761 . . . 4 (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴𝑛)
21biimpi 120 . . 3 (𝐴 ∈ Fin → ∃𝑛 ∈ ω 𝐴𝑛)
32adantl 277 . 2 ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑛 ∈ ω 𝐴𝑛)
4 simprlr 538 . . . 4 ((𝑛 ∈ ω ∧ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴𝑛)) → 𝐴 ∈ Fin)
5 breq2 4008 . . . . . . . . 9 (𝑤 = ∅ → (𝑥𝑤𝑥 ≈ ∅))
65anbi2d 464 . . . . . . . 8 (𝑤 = ∅ → (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑤) ↔ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ ∅)))
76imbi1d 231 . . . . . . 7 (𝑤 = ∅ → ((((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑤) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)) ↔ (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ ∅) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))))
87albidv 1824 . . . . . 6 (𝑤 = ∅ → (∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑤) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)) ↔ ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ ∅) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))))
9 breq2 4008 . . . . . . . . 9 (𝑤 = 𝑘 → (𝑥𝑤𝑥𝑘))
109anbi2d 464 . . . . . . . 8 (𝑤 = 𝑘 → (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑤) ↔ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑘)))
1110imbi1d 231 . . . . . . 7 (𝑤 = 𝑘 → ((((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑤) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)) ↔ (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))))
1211albidv 1824 . . . . . 6 (𝑤 = 𝑘 → (∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑤) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)) ↔ ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))))
13 breq2 4008 . . . . . . . . 9 (𝑤 = suc 𝑘 → (𝑥𝑤𝑥 ≈ suc 𝑘))
1413anbi2d 464 . . . . . . . 8 (𝑤 = suc 𝑘 → (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑤) ↔ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)))
1514imbi1d 231 . . . . . . 7 (𝑤 = suc 𝑘 → ((((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑤) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)) ↔ (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))))
1615albidv 1824 . . . . . 6 (𝑤 = suc 𝑘 → (∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑤) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)) ↔ ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))))
17 breq2 4008 . . . . . . . . 9 (𝑤 = 𝑛 → (𝑥𝑤𝑥𝑛))
1817anbi2d 464 . . . . . . . 8 (𝑤 = 𝑛 → (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑤) ↔ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑛)))
1918imbi1d 231 . . . . . . 7 (𝑤 = 𝑛 → ((((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑤) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)) ↔ (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑛) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))))
2019albidv 1824 . . . . . 6 (𝑤 = 𝑛 → (∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑤) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)) ↔ ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑛) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))))
21 iso0 5818 . . . . . . . . . 10 ∅ Isom < , < (∅, ∅)
22 en0 6795 . . . . . . . . . . . . . . . . 17 (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
2322biimpi 120 . . . . . . . . . . . . . . . 16 (𝑥 ≈ ∅ → 𝑥 = ∅)
2423fveq2d 5520 . . . . . . . . . . . . . . 15 (𝑥 ≈ ∅ → (♯‘𝑥) = (♯‘∅))
25 hash0 10776 . . . . . . . . . . . . . . 15 (♯‘∅) = 0
2624, 25eqtrdi 2226 . . . . . . . . . . . . . 14 (𝑥 ≈ ∅ → (♯‘𝑥) = 0)
2726oveq2d 5891 . . . . . . . . . . . . 13 (𝑥 ≈ ∅ → (1...(♯‘𝑥)) = (1...0))
28 fz10 10046 . . . . . . . . . . . . 13 (1...0) = ∅
2927, 28eqtrdi 2226 . . . . . . . . . . . 12 (𝑥 ≈ ∅ → (1...(♯‘𝑥)) = ∅)
30 isoeq4 5805 . . . . . . . . . . . 12 ((1...(♯‘𝑥)) = ∅ → (∅ Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ ∅ Isom < , < (∅, 𝑥)))
3129, 30syl 14 . . . . . . . . . . 11 (𝑥 ≈ ∅ → (∅ Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ ∅ Isom < , < (∅, 𝑥)))
32 isoeq5 5806 . . . . . . . . . . . 12 (𝑥 = ∅ → (∅ Isom < , < (∅, 𝑥) ↔ ∅ Isom < , < (∅, ∅)))
3323, 32syl 14 . . . . . . . . . . 11 (𝑥 ≈ ∅ → (∅ Isom < , < (∅, 𝑥) ↔ ∅ Isom < , < (∅, ∅)))
3431, 33bitrd 188 . . . . . . . . . 10 (𝑥 ≈ ∅ → (∅ Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ ∅ Isom < , < (∅, ∅)))
3521, 34mpbiri 168 . . . . . . . . 9 (𝑥 ≈ ∅ → ∅ Isom < , < ((1...(♯‘𝑥)), 𝑥))
36 0ex 4131 . . . . . . . . . 10 ∅ ∈ V
37 isoeq1 5802 . . . . . . . . . 10 (𝑓 = ∅ → (𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ ∅ Isom < , < ((1...(♯‘𝑥)), 𝑥)))
3836, 37spcev 2833 . . . . . . . . 9 (∅ Isom < , < ((1...(♯‘𝑥)), 𝑥) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))
3935, 38syl 14 . . . . . . . 8 (𝑥 ≈ ∅ → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))
4039adantl 277 . . . . . . 7 (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ ∅) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))
4140ax-gen 1449 . . . . . 6 𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ ∅) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))
42 sseq1 3179 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑎 ⊆ ℤ ↔ 𝑥 ⊆ ℤ))
43 eleq1 2240 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑎 ∈ Fin ↔ 𝑥 ∈ Fin))
4442, 43anbi12d 473 . . . . . . . . . 10 (𝑎 = 𝑥 → ((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ↔ (𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin)))
45 breq1 4007 . . . . . . . . . 10 (𝑎 = 𝑥 → (𝑎𝑘𝑥𝑘))
4644, 45anbi12d 473 . . . . . . . . 9 (𝑎 = 𝑥 → (((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) ↔ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑘)))
47 fveq2 5516 . . . . . . . . . . . . 13 (𝑎 = 𝑥 → (♯‘𝑎) = (♯‘𝑥))
4847oveq2d 5891 . . . . . . . . . . . 12 (𝑎 = 𝑥 → (1...(♯‘𝑎)) = (1...(♯‘𝑥)))
49 isoeq4 5805 . . . . . . . . . . . 12 ((1...(♯‘𝑎)) = (1...(♯‘𝑥)) → (𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎) ↔ 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑎)))
5048, 49syl 14 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎) ↔ 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑎)))
51 isoeq5 5806 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑎) ↔ 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)))
5250, 51bitrd 188 . . . . . . . . . 10 (𝑎 = 𝑥 → (𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎) ↔ 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)))
5352exbidv 1825 . . . . . . . . 9 (𝑎 = 𝑥 → (∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎) ↔ ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)))
5446, 53imbi12d 234 . . . . . . . 8 (𝑎 = 𝑥 → ((((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎)) ↔ (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))))
5554cbvalv 1917 . . . . . . 7 (∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎)) ↔ ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)))
56 simprll 537 . . . . . . . . . . . . 13 (((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) → 𝑥 ⊆ ℤ)
57 zssq 9627 . . . . . . . . . . . . 13 ℤ ⊆ ℚ
5856, 57sstrdi 3168 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) → 𝑥 ⊆ ℚ)
59 simprlr 538 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) → 𝑥 ∈ Fin)
60 vex 2741 . . . . . . . . . . . . . . . 16 𝑘 ∈ V
61 nsuceq0g 4419 . . . . . . . . . . . . . . . 16 (𝑘 ∈ V → suc 𝑘 ≠ ∅)
6260, 61ax-mp 5 . . . . . . . . . . . . . . 15 suc 𝑘 ≠ ∅
6362neii 2349 . . . . . . . . . . . . . 14 ¬ suc 𝑘 = ∅
64 simplrr 536 . . . . . . . . . . . . . . . . . 18 ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ 𝑥 = ∅) → 𝑥 ≈ suc 𝑘)
6564ensymd 6783 . . . . . . . . . . . . . . . . 17 ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ 𝑥 = ∅) → suc 𝑘𝑥)
66 simpr 110 . . . . . . . . . . . . . . . . 17 ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ 𝑥 = ∅) → 𝑥 = ∅)
6765, 66breqtrd 4030 . . . . . . . . . . . . . . . 16 ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ 𝑥 = ∅) → suc 𝑘 ≈ ∅)
68 en0 6795 . . . . . . . . . . . . . . . 16 (suc 𝑘 ≈ ∅ ↔ suc 𝑘 = ∅)
6967, 68sylib 122 . . . . . . . . . . . . . . 15 ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ 𝑥 = ∅) → suc 𝑘 = ∅)
7069ex 115 . . . . . . . . . . . . . 14 (((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) → (𝑥 = ∅ → suc 𝑘 = ∅))
7163, 70mtoi 664 . . . . . . . . . . . . 13 (((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) → ¬ 𝑥 = ∅)
7271neqned 2354 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) → 𝑥 ≠ ∅)
73 fimaxq 10807 . . . . . . . . . . . 12 ((𝑥 ⊆ ℚ ∧ 𝑥 ∈ Fin ∧ 𝑥 ≠ ∅) → ∃𝑚𝑥𝑧𝑥 𝑧𝑚)
7458, 59, 72, 73syl3anc 1238 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) → ∃𝑚𝑥𝑧𝑥 𝑧𝑚)
75 simplll 533 . . . . . . . . . . . 12 ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ (𝑚𝑥 ∧ ∀𝑧𝑥 𝑧𝑚)) → 𝑘 ∈ ω)
76 simpllr 534 . . . . . . . . . . . 12 ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ (𝑚𝑥 ∧ ∀𝑧𝑥 𝑧𝑚)) → ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎)))
7756adantr 276 . . . . . . . . . . . 12 ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ (𝑚𝑥 ∧ ∀𝑧𝑥 𝑧𝑚)) → 𝑥 ⊆ ℤ)
7859adantr 276 . . . . . . . . . . . 12 ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ (𝑚𝑥 ∧ ∀𝑧𝑥 𝑧𝑚)) → 𝑥 ∈ Fin)
79 simplrr 536 . . . . . . . . . . . 12 ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ (𝑚𝑥 ∧ ∀𝑧𝑥 𝑧𝑚)) → 𝑥 ≈ suc 𝑘)
80 simprl 529 . . . . . . . . . . . 12 ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ (𝑚𝑥 ∧ ∀𝑧𝑥 𝑧𝑚)) → 𝑚𝑥)
81 simprr 531 . . . . . . . . . . . 12 ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ (𝑚𝑥 ∧ ∀𝑧𝑥 𝑧𝑚)) → ∀𝑧𝑥 𝑧𝑚)
8275, 76, 77, 78, 79, 80, 81zfz1isolem1 10820 . . . . . . . . . . 11 ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ (𝑚𝑥 ∧ ∀𝑧𝑥 𝑧𝑚)) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))
8374, 82rexlimddv 2599 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))
8483ex 115 . . . . . . . . 9 ((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) → (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)))
8584alrimiv 1874 . . . . . . . 8 ((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) → ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)))
8685ex 115 . . . . . . 7 (𝑘 ∈ ω → (∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎)) → ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))))
8755, 86biimtrrid 153 . . . . . 6 (𝑘 ∈ ω → (∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)) → ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))))
888, 12, 16, 20, 41, 87finds 4600 . . . . 5 (𝑛 ∈ ω → ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑛) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)))
8988adantr 276 . . . 4 ((𝑛 ∈ ω ∧ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴𝑛)) → ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑛) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)))
90 simpr 110 . . . 4 ((𝑛 ∈ ω ∧ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴𝑛)) → ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴𝑛))
91 sseq1 3179 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 ⊆ ℤ ↔ 𝐴 ⊆ ℤ))
92 eleq1 2240 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 ∈ Fin ↔ 𝐴 ∈ Fin))
9391, 92anbi12d 473 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ↔ (𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin)))
94 breq1 4007 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝑛𝐴𝑛))
9593, 94anbi12d 473 . . . . . 6 (𝑥 = 𝐴 → (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑛) ↔ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴𝑛)))
96 fveq2 5516 . . . . . . . . . 10 (𝑥 = 𝐴 → (♯‘𝑥) = (♯‘𝐴))
9796oveq2d 5891 . . . . . . . . 9 (𝑥 = 𝐴 → (1...(♯‘𝑥)) = (1...(♯‘𝐴)))
98 isoeq4 5805 . . . . . . . . 9 ((1...(♯‘𝑥)) = (1...(♯‘𝐴)) → (𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝑥)))
9997, 98syl 14 . . . . . . . 8 (𝑥 = 𝐴 → (𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝑥)))
100 isoeq5 5806 . . . . . . . 8 (𝑥 = 𝐴 → (𝑓 Isom < , < ((1...(♯‘𝐴)), 𝑥) ↔ 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴)))
10199, 100bitrd 188 . . . . . . 7 (𝑥 = 𝐴 → (𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴)))
102101exbidv 1825 . . . . . 6 (𝑥 = 𝐴 → (∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴)))
10395, 102imbi12d 234 . . . . 5 (𝑥 = 𝐴 → ((((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑛) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)) ↔ (((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴𝑛) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴))))
104103spcgv 2825 . . . 4 (𝐴 ∈ Fin → (∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑛) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)) → (((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴𝑛) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴))))
1054, 89, 90, 104syl3c 63 . . 3 ((𝑛 ∈ ω ∧ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴𝑛)) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴))
106105an12s 565 . 2 (((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴))
1073, 106rexlimddv 2599 1 ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351   = wceq 1353  wex 1492  wcel 2148  wne 2347  wral 2455  wrex 2456  Vcvv 2738  wss 3130  c0 3423   class class class wbr 4004  suc csuc 4366  ωcom 4590  cfv 5217   Isom wiso 5218  (class class class)co 5875  cen 6738  Fincfn 6740  0cc0 7811  1c1 7812   < clt 7992  cle 7993  cz 9253  cq 9619  ...cfz 10008  chash 10755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-isom 5226  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-frec 6392  df-1o 6417  df-oadd 6421  df-er 6535  df-en 6741  df-dom 6742  df-fin 6743  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-n0 9177  df-z 9254  df-uz 9529  df-q 9620  df-rp 9654  df-fz 10009  df-ihash 10756
This theorem is referenced by:  summodclem2  11390  zsumdc  11392  prodmodclem2  11585  zproddc  11587
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