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Theorem zfz1isolem1 10241
Description: Lemma for zfz1iso 10242. Existence of an order isomorphism given the existence of shorter isomorphisms. (Contributed by Jim Kingdon, 7-Sep-2022.)
Hypotheses
Ref Expression
zfz1isolem1.k (𝜑𝐾 ∈ ω)
zfz1isolem1.h (𝜑 → ∀𝑦(((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑦𝐾) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦)))
zfz1isolem1.xz (𝜑𝑋 ⊆ ℤ)
zfz1isolem1.xf (𝜑𝑋 ∈ Fin)
zfz1isolem1.xs (𝜑𝑋 ≈ suc 𝐾)
zfz1isolem1.mx (𝜑𝑀𝑋)
zfz1isolem1.m (𝜑 → ∀𝑧𝑋 𝑧𝑀)
Assertion
Ref Expression
zfz1isolem1 (𝜑 → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑋)), 𝑋))
Distinct variable groups:   𝑦,𝐾   𝑧,𝑀   𝑓,𝑀,𝑦   𝑧,𝑋   𝑓,𝑋,𝑦
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑓)   𝐾(𝑧,𝑓)

Proof of Theorem zfz1isolem1
Dummy variables 𝑎 𝑏 𝑔 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zfz1isolem1.xz . . . . . 6 (𝜑𝑋 ⊆ ℤ)
21ssdifssd 3138 . . . . 5 (𝜑 → (𝑋 ∖ {𝑀}) ⊆ ℤ)
3 zfz1isolem1.xf . . . . . 6 (𝜑𝑋 ∈ Fin)
4 zfz1isolem1.mx . . . . . 6 (𝜑𝑀𝑋)
5 diffisn 6607 . . . . . 6 ((𝑋 ∈ Fin ∧ 𝑀𝑋) → (𝑋 ∖ {𝑀}) ∈ Fin)
63, 4, 5syl2anc 403 . . . . 5 (𝜑 → (𝑋 ∖ {𝑀}) ∈ Fin)
7 zfz1isolem1.k . . . . . 6 (𝜑𝐾 ∈ ω)
8 zfz1isolem1.xs . . . . . 6 (𝜑𝑋 ≈ suc 𝐾)
9 dif1en 6593 . . . . . 6 ((𝐾 ∈ ω ∧ 𝑋 ≈ suc 𝐾𝑀𝑋) → (𝑋 ∖ {𝑀}) ≈ 𝐾)
107, 8, 4, 9syl3anc 1174 . . . . 5 (𝜑 → (𝑋 ∖ {𝑀}) ≈ 𝐾)
112, 6, 10jca31 302 . . . 4 (𝜑 → (((𝑋 ∖ {𝑀}) ⊆ ℤ ∧ (𝑋 ∖ {𝑀}) ∈ Fin) ∧ (𝑋 ∖ {𝑀}) ≈ 𝐾))
12 zfz1isolem1.h . . . . 5 (𝜑 → ∀𝑦(((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑦𝐾) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦)))
13 sseq1 3047 . . . . . . . . . 10 (𝑦 = (𝑋 ∖ {𝑀}) → (𝑦 ⊆ ℤ ↔ (𝑋 ∖ {𝑀}) ⊆ ℤ))
14 eleq1 2150 . . . . . . . . . 10 (𝑦 = (𝑋 ∖ {𝑀}) → (𝑦 ∈ Fin ↔ (𝑋 ∖ {𝑀}) ∈ Fin))
1513, 14anbi12d 457 . . . . . . . . 9 (𝑦 = (𝑋 ∖ {𝑀}) → ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ↔ ((𝑋 ∖ {𝑀}) ⊆ ℤ ∧ (𝑋 ∖ {𝑀}) ∈ Fin)))
16 breq1 3848 . . . . . . . . 9 (𝑦 = (𝑋 ∖ {𝑀}) → (𝑦𝐾 ↔ (𝑋 ∖ {𝑀}) ≈ 𝐾))
1715, 16anbi12d 457 . . . . . . . 8 (𝑦 = (𝑋 ∖ {𝑀}) → (((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑦𝐾) ↔ (((𝑋 ∖ {𝑀}) ⊆ ℤ ∧ (𝑋 ∖ {𝑀}) ∈ Fin) ∧ (𝑋 ∖ {𝑀}) ≈ 𝐾)))
18 fveq2 5305 . . . . . . . . . . . 12 (𝑦 = (𝑋 ∖ {𝑀}) → (♯‘𝑦) = (♯‘(𝑋 ∖ {𝑀})))
1918oveq2d 5668 . . . . . . . . . . 11 (𝑦 = (𝑋 ∖ {𝑀}) → (1...(♯‘𝑦)) = (1...(♯‘(𝑋 ∖ {𝑀}))))
20 isoeq4 5583 . . . . . . . . . . 11 ((1...(♯‘𝑦)) = (1...(♯‘(𝑋 ∖ {𝑀}))) → (𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦) ↔ 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), 𝑦)))
2119, 20syl 14 . . . . . . . . . 10 (𝑦 = (𝑋 ∖ {𝑀}) → (𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦) ↔ 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), 𝑦)))
22 isoeq5 5584 . . . . . . . . . 10 (𝑦 = (𝑋 ∖ {𝑀}) → (𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), 𝑦) ↔ 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))))
2321, 22bitrd 186 . . . . . . . . 9 (𝑦 = (𝑋 ∖ {𝑀}) → (𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦) ↔ 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))))
2423exbidv 1753 . . . . . . . 8 (𝑦 = (𝑋 ∖ {𝑀}) → (∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦) ↔ ∃𝑓 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))))
2517, 24imbi12d 232 . . . . . . 7 (𝑦 = (𝑋 ∖ {𝑀}) → ((((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑦𝐾) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦)) ↔ ((((𝑋 ∖ {𝑀}) ⊆ ℤ ∧ (𝑋 ∖ {𝑀}) ∈ Fin) ∧ (𝑋 ∖ {𝑀}) ≈ 𝐾) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))))
2625spcgv 2706 . . . . . 6 ((𝑋 ∖ {𝑀}) ∈ Fin → (∀𝑦(((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑦𝐾) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦)) → ((((𝑋 ∖ {𝑀}) ⊆ ℤ ∧ (𝑋 ∖ {𝑀}) ∈ Fin) ∧ (𝑋 ∖ {𝑀}) ≈ 𝐾) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))))
276, 26syl 14 . . . . 5 (𝜑 → (∀𝑦(((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑦𝐾) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦)) → ((((𝑋 ∖ {𝑀}) ⊆ ℤ ∧ (𝑋 ∖ {𝑀}) ∈ Fin) ∧ (𝑋 ∖ {𝑀}) ≈ 𝐾) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))))
2812, 27mpd 13 . . . 4 (𝜑 → ((((𝑋 ∖ {𝑀}) ⊆ ℤ ∧ (𝑋 ∖ {𝑀}) ∈ Fin) ∧ (𝑋 ∖ {𝑀}) ≈ 𝐾) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))))
2911, 28mpd 13 . . 3 (𝜑 → ∃𝑓 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))
30 isoeq1 5580 . . . 4 (𝑓 = 𝑔 → (𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})) ↔ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))))
3130cbvexv 1843 . . 3 (∃𝑓 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})) ↔ ∃𝑔 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))
3229, 31sylib 120 . 2 (𝜑 → ∃𝑔 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))
33 df-isom 5024 . . . . . . . . 9 (𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})) ↔ (𝑔:(1...(♯‘(𝑋 ∖ {𝑀})))–1-1-onto→(𝑋 ∖ {𝑀}) ∧ ∀𝑢 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))∀𝑣 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))(𝑢 < 𝑣 ↔ (𝑔𝑢) < (𝑔𝑣))))
3433biimpi 118 . . . . . . . 8 (𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})) → (𝑔:(1...(♯‘(𝑋 ∖ {𝑀})))–1-1-onto→(𝑋 ∖ {𝑀}) ∧ ∀𝑢 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))∀𝑣 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))(𝑢 < 𝑣 ↔ (𝑔𝑢) < (𝑔𝑣))))
3534adantl 271 . . . . . . 7 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → (𝑔:(1...(♯‘(𝑋 ∖ {𝑀})))–1-1-onto→(𝑋 ∖ {𝑀}) ∧ ∀𝑢 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))∀𝑣 ∈ (1...(♯‘(𝑋 ∖ {𝑀})))(𝑢 < 𝑣 ↔ (𝑔𝑢) < (𝑔𝑣))))
3635simpld 110 . . . . . 6 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → 𝑔:(1...(♯‘(𝑋 ∖ {𝑀})))–1-1-onto→(𝑋 ∖ {𝑀}))
37 hashcl 10185 . . . . . . . . 9 (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ0)
383, 37syl 14 . . . . . . . 8 (𝜑 → (♯‘𝑋) ∈ ℕ0)
3938adantr 270 . . . . . . 7 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → (♯‘𝑋) ∈ ℕ0)
404adantr 270 . . . . . . 7 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → 𝑀𝑋)
41 f1osng 5294 . . . . . . 7 (((♯‘𝑋) ∈ ℕ0𝑀𝑋) → {⟨(♯‘𝑋), 𝑀⟩}:{(♯‘𝑋)}–1-1-onto→{𝑀})
4239, 40, 41syl2anc 403 . . . . . 6 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → {⟨(♯‘𝑋), 𝑀⟩}:{(♯‘𝑋)}–1-1-onto→{𝑀})
43 fzp1disj 9490 . . . . . . . 8 ((1...(♯‘(𝑋 ∖ {𝑀}))) ∩ {((♯‘(𝑋 ∖ {𝑀})) + 1)}) = ∅
44 hashdifsn 10223 . . . . . . . . . . . . 13 ((𝑋 ∈ Fin ∧ 𝑀𝑋) → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1))
453, 4, 44syl2anc 403 . . . . . . . . . . . 12 (𝜑 → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1))
4645oveq1d 5667 . . . . . . . . . . 11 (𝜑 → ((♯‘(𝑋 ∖ {𝑀})) + 1) = (((♯‘𝑋) − 1) + 1))
4738nn0cnd 8726 . . . . . . . . . . . 12 (𝜑 → (♯‘𝑋) ∈ ℂ)
48 1cnd 7502 . . . . . . . . . . . 12 (𝜑 → 1 ∈ ℂ)
4947, 48npcand 7795 . . . . . . . . . . 11 (𝜑 → (((♯‘𝑋) − 1) + 1) = (♯‘𝑋))
5046, 49eqtrd 2120 . . . . . . . . . 10 (𝜑 → ((♯‘(𝑋 ∖ {𝑀})) + 1) = (♯‘𝑋))
5150sneqd 3459 . . . . . . . . 9 (𝜑 → {((♯‘(𝑋 ∖ {𝑀})) + 1)} = {(♯‘𝑋)})
5251ineq2d 3201 . . . . . . . 8 (𝜑 → ((1...(♯‘(𝑋 ∖ {𝑀}))) ∩ {((♯‘(𝑋 ∖ {𝑀})) + 1)}) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∩ {(♯‘𝑋)}))
5343, 52syl5reqr 2135 . . . . . . 7 (𝜑 → ((1...(♯‘(𝑋 ∖ {𝑀}))) ∩ {(♯‘𝑋)}) = ∅)
5453adantr 270 . . . . . 6 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → ((1...(♯‘(𝑋 ∖ {𝑀}))) ∩ {(♯‘𝑋)}) = ∅)
55 incom 3192 . . . . . . . 8 ((𝑋 ∖ {𝑀}) ∩ {𝑀}) = ({𝑀} ∩ (𝑋 ∖ {𝑀}))
56 disjdif 3355 . . . . . . . 8 ({𝑀} ∩ (𝑋 ∖ {𝑀})) = ∅
5755, 56eqtri 2108 . . . . . . 7 ((𝑋 ∖ {𝑀}) ∩ {𝑀}) = ∅
5857a1i 9 . . . . . 6 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → ((𝑋 ∖ {𝑀}) ∩ {𝑀}) = ∅)
59 f1oun 5273 . . . . . 6 (((𝑔:(1...(♯‘(𝑋 ∖ {𝑀})))–1-1-onto→(𝑋 ∖ {𝑀}) ∧ {⟨(♯‘𝑋), 𝑀⟩}:{(♯‘𝑋)}–1-1-onto→{𝑀}) ∧ (((1...(♯‘(𝑋 ∖ {𝑀}))) ∩ {(♯‘𝑋)}) = ∅ ∧ ((𝑋 ∖ {𝑀}) ∩ {𝑀}) = ∅)) → (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)})–1-1-onto→((𝑋 ∖ {𝑀}) ∪ {𝑀}))
6036, 42, 54, 58, 59syl22anc 1175 . . . . 5 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)})–1-1-onto→((𝑋 ∖ {𝑀}) ∪ {𝑀}))
613, 4zfz1isolemsplit 10239 . . . . . . 7 (𝜑 → (1...(♯‘𝑋)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)}))
62 fidifsnid 6585 . . . . . . . . 9 ((𝑋 ∈ Fin ∧ 𝑀𝑋) → ((𝑋 ∖ {𝑀}) ∪ {𝑀}) = 𝑋)
633, 4, 62syl2anc 403 . . . . . . . 8 (𝜑 → ((𝑋 ∖ {𝑀}) ∪ {𝑀}) = 𝑋)
6463eqcomd 2093 . . . . . . 7 (𝜑𝑋 = ((𝑋 ∖ {𝑀}) ∪ {𝑀}))
65 f1oeq23 5247 . . . . . . 7 (((1...(♯‘𝑋)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)}) ∧ 𝑋 = ((𝑋 ∖ {𝑀}) ∪ {𝑀})) → ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):(1...(♯‘𝑋))–1-1-onto𝑋 ↔ (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)})–1-1-onto→((𝑋 ∖ {𝑀}) ∪ {𝑀})))
6661, 64, 65syl2anc 403 . . . . . 6 (𝜑 → ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):(1...(♯‘𝑋))–1-1-onto𝑋 ↔ (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)})–1-1-onto→((𝑋 ∖ {𝑀}) ∪ {𝑀})))
6766adantr 270 . . . . 5 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):(1...(♯‘𝑋))–1-1-onto𝑋 ↔ (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)})–1-1-onto→((𝑋 ∖ {𝑀}) ∪ {𝑀})))
6860, 67mpbird 165 . . . 4 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):(1...(♯‘𝑋))–1-1-onto𝑋)
693ad2antrr 472 . . . . . 6 (((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → 𝑋 ∈ Fin)
701ad2antrr 472 . . . . . 6 (((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → 𝑋 ⊆ ℤ)
714ad2antrr 472 . . . . . 6 (((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → 𝑀𝑋)
72 zfz1isolem1.m . . . . . . 7 (𝜑 → ∀𝑧𝑋 𝑧𝑀)
7372ad2antrr 472 . . . . . 6 (((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → ∀𝑧𝑋 𝑧𝑀)
74 simplr 497 . . . . . 6 (((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))
75 simprl 498 . . . . . 6 (((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → 𝑎 ∈ (1...(♯‘𝑋)))
76 simprr 499 . . . . . 6 (((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → 𝑏 ∈ (1...(♯‘𝑋)))
7769, 70, 71, 73, 74, 75, 76zfz1isolemiso 10240 . . . . 5 (((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → (𝑎 < 𝑏 ↔ ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝑎) < ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝑏)))
7877ralrimivva 2455 . . . 4 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → ∀𝑎 ∈ (1...(♯‘𝑋))∀𝑏 ∈ (1...(♯‘𝑋))(𝑎 < 𝑏 ↔ ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝑎) < ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝑏)))
79 df-isom 5024 . . . 4 ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) Isom < , < ((1...(♯‘𝑋)), 𝑋) ↔ ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):(1...(♯‘𝑋))–1-1-onto𝑋 ∧ ∀𝑎 ∈ (1...(♯‘𝑋))∀𝑏 ∈ (1...(♯‘𝑋))(𝑎 < 𝑏 ↔ ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝑎) < ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝑏))))
8068, 78, 79sylanbrc 408 . . 3 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) Isom < , < ((1...(♯‘𝑋)), 𝑋))
81 vex 2622 . . . . . . 7 𝑔 ∈ V
8281a1i 9 . . . . . 6 (𝜑𝑔 ∈ V)
83 opexg 4055 . . . . . . . 8 (((♯‘𝑋) ∈ ℕ0𝑀𝑋) → ⟨(♯‘𝑋), 𝑀⟩ ∈ V)
8438, 4, 83syl2anc 403 . . . . . . 7 (𝜑 → ⟨(♯‘𝑋), 𝑀⟩ ∈ V)
85 snexg 4019 . . . . . . 7 (⟨(♯‘𝑋), 𝑀⟩ ∈ V → {⟨(♯‘𝑋), 𝑀⟩} ∈ V)
8684, 85syl 14 . . . . . 6 (𝜑 → {⟨(♯‘𝑋), 𝑀⟩} ∈ V)
87 unexg 4268 . . . . . 6 ((𝑔 ∈ V ∧ {⟨(♯‘𝑋), 𝑀⟩} ∈ V) → (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) ∈ V)
8882, 86, 87syl2anc 403 . . . . 5 (𝜑 → (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) ∈ V)
89 isoeq1 5580 . . . . . 6 (𝑓 = (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) → (𝑓 Isom < , < ((1...(♯‘𝑋)), 𝑋) ↔ (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) Isom < , < ((1...(♯‘𝑋)), 𝑋)))
9089spcegv 2707 . . . . 5 ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) ∈ V → ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) Isom < , < ((1...(♯‘𝑋)), 𝑋) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑋)), 𝑋)))
9188, 90syl 14 . . . 4 (𝜑 → ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) Isom < , < ((1...(♯‘𝑋)), 𝑋) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑋)), 𝑋)))
9291adantr 270 . . 3 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) Isom < , < ((1...(♯‘𝑋)), 𝑋) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑋)), 𝑋)))
9380, 92mpd 13 . 2 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑋)), 𝑋))
9432, 93exlimddv 1826 1 (𝜑 → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑋)), 𝑋))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1287   = wceq 1289  wex 1426  wcel 1438  wral 2359  Vcvv 2619  cdif 2996  cun 2997  cin 2998  wss 2999  c0 3286  {csn 3446  cop 3449   class class class wbr 3845  suc csuc 4192  ωcom 4405  1-1-ontowf1o 5014  cfv 5015   Isom wiso 5016  (class class class)co 5652  cen 6453  Fincfn 6455  1c1 7349   + caddc 7351   < clt 7520  cle 7521  cmin 7651  0cn0 8671  cz 8748  ...cfz 9422  chash 10179
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 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-addcom 7443  ax-addass 7445  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-0id 7451  ax-rnegex 7452  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459
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-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-isom 5024  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-frec 6156  df-1o 6181  df-oadd 6185  df-er 6290  df-en 6456  df-dom 6457  df-fin 6458  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-inn 8421  df-n0 8672  df-z 8749  df-uz 9018  df-fz 9423  df-ihash 10180
This theorem is referenced by:  zfz1iso  10242
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