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Theorem zfz1iso 10211
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 6458 . . . 4 (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴𝑛)
21biimpi 118 . . 3 (𝐴 ∈ Fin → ∃𝑛 ∈ ω 𝐴𝑛)
32adantl 271 . 2 ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑛 ∈ ω 𝐴𝑛)
4 simprlr 505 . . . 4 ((𝑛 ∈ ω ∧ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴𝑛)) → 𝐴 ∈ Fin)
5 breq2 3841 . . . . . . . . 9 (𝑤 = ∅ → (𝑥𝑤𝑥 ≈ ∅))
65anbi2d 452 . . . . . . . 8 (𝑤 = ∅ → (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑤) ↔ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ ∅)))
76imbi1d 229 . . . . . . 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 1752 . . . . . 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 3841 . . . . . . . . 9 (𝑤 = 𝑘 → (𝑥𝑤𝑥𝑘))
109anbi2d 452 . . . . . . . 8 (𝑤 = 𝑘 → (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑤) ↔ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑘)))
1110imbi1d 229 . . . . . . 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 1752 . . . . . 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 3841 . . . . . . . . 9 (𝑤 = suc 𝑘 → (𝑥𝑤𝑥 ≈ suc 𝑘))
1413anbi2d 452 . . . . . . . 8 (𝑤 = suc 𝑘 → (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑤) ↔ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)))
1514imbi1d 229 . . . . . . 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 1752 . . . . . 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 3841 . . . . . . . . 9 (𝑤 = 𝑛 → (𝑥𝑤𝑥𝑛))
1817anbi2d 452 . . . . . . . 8 (𝑤 = 𝑛 → (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑤) ↔ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑛)))
1918imbi1d 229 . . . . . . 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 1752 . . . . . 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 5578 . . . . . . . . . 10 ∅ Isom < , < (∅, ∅)
22 en0 6492 . . . . . . . . . . . . . . . . 17 (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
2322biimpi 118 . . . . . . . . . . . . . . . 16 (𝑥 ≈ ∅ → 𝑥 = ∅)
2423fveq2d 5293 . . . . . . . . . . . . . . 15 (𝑥 ≈ ∅ → (♯‘𝑥) = (♯‘∅))
25 hash0 10170 . . . . . . . . . . . . . . 15 (♯‘∅) = 0
2624, 25syl6eq 2136 . . . . . . . . . . . . . 14 (𝑥 ≈ ∅ → (♯‘𝑥) = 0)
2726oveq2d 5650 . . . . . . . . . . . . 13 (𝑥 ≈ ∅ → (1...(♯‘𝑥)) = (1...0))
28 fz10 9429 . . . . . . . . . . . . 13 (1...0) = ∅
2927, 28syl6eq 2136 . . . . . . . . . . . 12 (𝑥 ≈ ∅ → (1...(♯‘𝑥)) = ∅)
30 isoeq4 5565 . . . . . . . . . . . 12 ((1...(♯‘𝑥)) = ∅ → (∅ Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ ∅ Isom < , < (∅, 𝑥)))
3129, 30syl 14 . . . . . . . . . . 11 (𝑥 ≈ ∅ → (∅ Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ ∅ Isom < , < (∅, 𝑥)))
32 isoeq5 5566 . . . . . . . . . . . 12 (𝑥 = ∅ → (∅ Isom < , < (∅, 𝑥) ↔ ∅ Isom < , < (∅, ∅)))
3323, 32syl 14 . . . . . . . . . . 11 (𝑥 ≈ ∅ → (∅ Isom < , < (∅, 𝑥) ↔ ∅ Isom < , < (∅, ∅)))
3431, 33bitrd 186 . . . . . . . . . 10 (𝑥 ≈ ∅ → (∅ Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ ∅ Isom < , < (∅, ∅)))
3521, 34mpbiri 166 . . . . . . . . 9 (𝑥 ≈ ∅ → ∅ Isom < , < ((1...(♯‘𝑥)), 𝑥))
36 0ex 3958 . . . . . . . . . 10 ∅ ∈ V
37 isoeq1 5562 . . . . . . . . . 10 (𝑓 = ∅ → (𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ ∅ Isom < , < ((1...(♯‘𝑥)), 𝑥)))
3836, 37spcev 2713 . . . . . . . . 9 (∅ Isom < , < ((1...(♯‘𝑥)), 𝑥) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))
3935, 38syl 14 . . . . . . . 8 (𝑥 ≈ ∅ → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))
4039adantl 271 . . . . . . 7 (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ ∅) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))
4140ax-gen 1383 . . . . . 6 𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ ∅) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))
42 sseq1 3045 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑎 ⊆ ℤ ↔ 𝑥 ⊆ ℤ))
43 eleq1 2150 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑎 ∈ Fin ↔ 𝑥 ∈ Fin))
4442, 43anbi12d 457 . . . . . . . . . 10 (𝑎 = 𝑥 → ((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ↔ (𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin)))
45 breq1 3840 . . . . . . . . . 10 (𝑎 = 𝑥 → (𝑎𝑘𝑥𝑘))
4644, 45anbi12d 457 . . . . . . . . 9 (𝑎 = 𝑥 → (((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) ↔ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑘)))
47 fveq2 5289 . . . . . . . . . . . . 13 (𝑎 = 𝑥 → (♯‘𝑎) = (♯‘𝑥))
4847oveq2d 5650 . . . . . . . . . . . 12 (𝑎 = 𝑥 → (1...(♯‘𝑎)) = (1...(♯‘𝑥)))
49 isoeq4 5565 . . . . . . . . . . . 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 5566 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑎) ↔ 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)))
5250, 51bitrd 186 . . . . . . . . . 10 (𝑎 = 𝑥 → (𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎) ↔ 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)))
5352exbidv 1753 . . . . . . . . 9 (𝑎 = 𝑥 → (∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎) ↔ ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)))
5446, 53imbi12d 232 . . . . . . . 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 1842 . . . . . . 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 504 . . . . . . . . . . . . 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 9081 . . . . . . . . . . . . 13 ℤ ⊆ ℚ
5856, 57syl6ss 3035 . . . . . . . . . . . 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 505 . . . . . . . . . . . 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 2622 . . . . . . . . . . . . . . . 16 𝑘 ∈ V
61 nsuceq0g 4236 . . . . . . . . . . . . . . . 16 (𝑘 ∈ V → suc 𝑘 ≠ ∅)
6260, 61ax-mp 7 . . . . . . . . . . . . . . 15 suc 𝑘 ≠ ∅
6362neii 2257 . . . . . . . . . . . . . 14 ¬ suc 𝑘 = ∅
64 simplrr 503 . . . . . . . . . . . . . . . . . 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 6480 . . . . . . . . . . . . . . . . 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 108 . . . . . . . . . . . . . . . . 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 3861 . . . . . . . . . . . . . . . 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 6492 . . . . . . . . . . . . . . . 16 (suc 𝑘 ≈ ∅ ↔ suc 𝑘 = ∅)
6967, 68sylib 120 . . . . . . . . . . . . . . 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 113 . . . . . . . . . . . . . 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 625 . . . . . . . . . . . . 13 (((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) → ¬ 𝑥 = ∅)
7271neqned 2262 . . . . . . . . . . . 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 10200 . . . . . . . . . . . 12 ((𝑥 ⊆ ℚ ∧ 𝑥 ∈ Fin ∧ 𝑥 ≠ ∅) → ∃𝑚𝑥𝑧𝑥 𝑧𝑚)
7458, 59, 72, 73syl3anc 1174 . . . . . . . . . . 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 500 . . . . . . . . . . . 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 501 . . . . . . . . . . . 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 270 . . . . . . . . . . . 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 270 . . . . . . . . . . . 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 503 . . . . . . . . . . . 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 498 . . . . . . . . . . . 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 499 . . . . . . . . . . . 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 10210 . . . . . . . . . . 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 2493 . . . . . . . . . 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 113 . . . . . . . . 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 1802 . . . . . . . 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 113 . . . . . . 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, 86syl5bir 151 . . . . . 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 4405 . . . . 5 (𝑛 ∈ ω → ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑛) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)))
8988adantr 270 . . . 4 ((𝑛 ∈ ω ∧ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴𝑛)) → ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑛) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)))
90 simpr 108 . . . 4 ((𝑛 ∈ ω ∧ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴𝑛)) → ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴𝑛))
91 sseq1 3045 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 ⊆ ℤ ↔ 𝐴 ⊆ ℤ))
92 eleq1 2150 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 ∈ Fin ↔ 𝐴 ∈ Fin))
9391, 92anbi12d 457 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ↔ (𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin)))
94 breq1 3840 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝑛𝐴𝑛))
9593, 94anbi12d 457 . . . . . 6 (𝑥 = 𝐴 → (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥𝑛) ↔ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴𝑛)))
96 fveq2 5289 . . . . . . . . . 10 (𝑥 = 𝐴 → (♯‘𝑥) = (♯‘𝐴))
9796oveq2d 5650 . . . . . . . . 9 (𝑥 = 𝐴 → (1...(♯‘𝑥)) = (1...(♯‘𝐴)))
98 isoeq4 5565 . . . . . . . . 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 5566 . . . . . . . 8 (𝑥 = 𝐴 → (𝑓 Isom < , < ((1...(♯‘𝐴)), 𝑥) ↔ 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴)))
10199, 100bitrd 186 . . . . . . 7 (𝑥 = 𝐴 → (𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴)))
102101exbidv 1753 . . . . . 6 (𝑥 = 𝐴 → (∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴)))
10395, 102imbi12d 232 . . . . 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 2706 . . . 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 62 . . 3 ((𝑛 ∈ ω ∧ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴𝑛)) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴))
106105an12s 532 . 2 (((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴))
1073, 106rexlimddv 2493 1 ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1287   = wceq 1289  wex 1426  wcel 1438  wne 2255  wral 2359  wrex 2360  Vcvv 2619  wss 2997  c0 3284   class class class wbr 3837  suc csuc 4183  ωcom 4395  cfv 5002   Isom wiso 5003  (class class class)co 5634  cen 6435  Fincfn 6437  0cc0 7329  1c1 7330   < clt 7501  cle 7502  cz 8720  cq 9073  ...cfz 9393  chash 10148
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-isom 5011  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-frec 6138  df-1o 6163  df-oadd 6167  df-er 6272  df-en 6438  df-dom 6439  df-fin 6440  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-fz 9394  df-ihash 10149
This theorem is referenced by:  isummolem2  10736  zisum  10738
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