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Theorem zfz1isolem1 10820
Description: Lemma for zfz1iso 10821. 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 3274 . . . . 5 (𝜑 → (𝑋 ∖ {𝑀}) ⊆ ℤ)
3 zfz1isolem1.xf . . . . . 6 (𝜑𝑋 ∈ Fin)
4 zfz1isolem1.mx . . . . . 6 (𝜑𝑀𝑋)
5 diffisn 6893 . . . . . 6 ((𝑋 ∈ Fin ∧ 𝑀𝑋) → (𝑋 ∖ {𝑀}) ∈ Fin)
63, 4, 5syl2anc 411 . . . . 5 (𝜑 → (𝑋 ∖ {𝑀}) ∈ Fin)
7 zfz1isolem1.k . . . . . 6 (𝜑𝐾 ∈ ω)
8 zfz1isolem1.xs . . . . . 6 (𝜑𝑋 ≈ suc 𝐾)
9 dif1en 6879 . . . . . 6 ((𝐾 ∈ ω ∧ 𝑋 ≈ suc 𝐾𝑀𝑋) → (𝑋 ∖ {𝑀}) ≈ 𝐾)
107, 8, 4, 9syl3anc 1238 . . . . 5 (𝜑 → (𝑋 ∖ {𝑀}) ≈ 𝐾)
112, 6, 10jca31 309 . . . 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 3179 . . . . . . . . . 10 (𝑦 = (𝑋 ∖ {𝑀}) → (𝑦 ⊆ ℤ ↔ (𝑋 ∖ {𝑀}) ⊆ ℤ))
14 eleq1 2240 . . . . . . . . . 10 (𝑦 = (𝑋 ∖ {𝑀}) → (𝑦 ∈ Fin ↔ (𝑋 ∖ {𝑀}) ∈ Fin))
1513, 14anbi12d 473 . . . . . . . . 9 (𝑦 = (𝑋 ∖ {𝑀}) → ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ↔ ((𝑋 ∖ {𝑀}) ⊆ ℤ ∧ (𝑋 ∖ {𝑀}) ∈ Fin)))
16 breq1 4007 . . . . . . . . 9 (𝑦 = (𝑋 ∖ {𝑀}) → (𝑦𝐾 ↔ (𝑋 ∖ {𝑀}) ≈ 𝐾))
1715, 16anbi12d 473 . . . . . . . 8 (𝑦 = (𝑋 ∖ {𝑀}) → (((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑦𝐾) ↔ (((𝑋 ∖ {𝑀}) ⊆ ℤ ∧ (𝑋 ∖ {𝑀}) ∈ Fin) ∧ (𝑋 ∖ {𝑀}) ≈ 𝐾)))
18 fveq2 5516 . . . . . . . . . . . 12 (𝑦 = (𝑋 ∖ {𝑀}) → (♯‘𝑦) = (♯‘(𝑋 ∖ {𝑀})))
1918oveq2d 5891 . . . . . . . . . . 11 (𝑦 = (𝑋 ∖ {𝑀}) → (1...(♯‘𝑦)) = (1...(♯‘(𝑋 ∖ {𝑀}))))
20 isoeq4 5805 . . . . . . . . . . 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 5806 . . . . . . . . . 10 (𝑦 = (𝑋 ∖ {𝑀}) → (𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), 𝑦) ↔ 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))))
2321, 22bitrd 188 . . . . . . . . 9 (𝑦 = (𝑋 ∖ {𝑀}) → (𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦) ↔ 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))))
2423exbidv 1825 . . . . . . . 8 (𝑦 = (𝑋 ∖ {𝑀}) → (∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦) ↔ ∃𝑓 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))))
2517, 24imbi12d 234 . . . . . . 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 2825 . . . . . 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 5802 . . . 4 (𝑓 = 𝑔 → (𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})) ↔ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))))
3130cbvexv 1918 . . 3 (∃𝑓 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})) ↔ ∃𝑔 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))
3229, 31sylib 122 . 2 (𝜑 → ∃𝑔 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))
33 df-isom 5226 . . . . . . . . 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 120 . . . . . . . 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 277 . . . . . . 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 112 . . . . . 6 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → 𝑔:(1...(♯‘(𝑋 ∖ {𝑀})))–1-1-onto→(𝑋 ∖ {𝑀}))
37 hashcl 10761 . . . . . . . . 9 (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ0)
383, 37syl 14 . . . . . . . 8 (𝜑 → (♯‘𝑋) ∈ ℕ0)
3938adantr 276 . . . . . . 7 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → (♯‘𝑋) ∈ ℕ0)
404adantr 276 . . . . . . 7 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → 𝑀𝑋)
41 f1osng 5503 . . . . . . 7 (((♯‘𝑋) ∈ ℕ0𝑀𝑋) → {⟨(♯‘𝑋), 𝑀⟩}:{(♯‘𝑋)}–1-1-onto→{𝑀})
4239, 40, 41syl2anc 411 . . . . . 6 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → {⟨(♯‘𝑋), 𝑀⟩}:{(♯‘𝑋)}–1-1-onto→{𝑀})
43 hashdifsn 10799 . . . . . . . . . . . . 13 ((𝑋 ∈ Fin ∧ 𝑀𝑋) → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1))
443, 4, 43syl2anc 411 . . . . . . . . . . . 12 (𝜑 → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1))
4544oveq1d 5890 . . . . . . . . . . 11 (𝜑 → ((♯‘(𝑋 ∖ {𝑀})) + 1) = (((♯‘𝑋) − 1) + 1))
4638nn0cnd 9231 . . . . . . . . . . . 12 (𝜑 → (♯‘𝑋) ∈ ℂ)
47 1cnd 7973 . . . . . . . . . . . 12 (𝜑 → 1 ∈ ℂ)
4846, 47npcand 8272 . . . . . . . . . . 11 (𝜑 → (((♯‘𝑋) − 1) + 1) = (♯‘𝑋))
4945, 48eqtrd 2210 . . . . . . . . . 10 (𝜑 → ((♯‘(𝑋 ∖ {𝑀})) + 1) = (♯‘𝑋))
5049sneqd 3606 . . . . . . . . 9 (𝜑 → {((♯‘(𝑋 ∖ {𝑀})) + 1)} = {(♯‘𝑋)})
5150ineq2d 3337 . . . . . . . 8 (𝜑 → ((1...(♯‘(𝑋 ∖ {𝑀}))) ∩ {((♯‘(𝑋 ∖ {𝑀})) + 1)}) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∩ {(♯‘𝑋)}))
52 fzp1disj 10080 . . . . . . . 8 ((1...(♯‘(𝑋 ∖ {𝑀}))) ∩ {((♯‘(𝑋 ∖ {𝑀})) + 1)}) = ∅
5351, 52eqtr3di 2225 . . . . . . 7 (𝜑 → ((1...(♯‘(𝑋 ∖ {𝑀}))) ∩ {(♯‘𝑋)}) = ∅)
5453adantr 276 . . . . . 6 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → ((1...(♯‘(𝑋 ∖ {𝑀}))) ∩ {(♯‘𝑋)}) = ∅)
55 incom 3328 . . . . . . . 8 ((𝑋 ∖ {𝑀}) ∩ {𝑀}) = ({𝑀} ∩ (𝑋 ∖ {𝑀}))
56 disjdif 3496 . . . . . . . 8 ({𝑀} ∩ (𝑋 ∖ {𝑀})) = ∅
5755, 56eqtri 2198 . . . . . . 7 ((𝑋 ∖ {𝑀}) ∩ {𝑀}) = ∅
5857a1i 9 . . . . . 6 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → ((𝑋 ∖ {𝑀}) ∩ {𝑀}) = ∅)
59 f1oun 5482 . . . . . 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 1239 . . . . 5 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)})–1-1-onto→((𝑋 ∖ {𝑀}) ∪ {𝑀}))
613, 4zfz1isolemsplit 10818 . . . . . . 7 (𝜑 → (1...(♯‘𝑋)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)}))
62 fidifsnid 6871 . . . . . . . . 9 ((𝑋 ∈ Fin ∧ 𝑀𝑋) → ((𝑋 ∖ {𝑀}) ∪ {𝑀}) = 𝑋)
633, 4, 62syl2anc 411 . . . . . . . 8 (𝜑 → ((𝑋 ∖ {𝑀}) ∪ {𝑀}) = 𝑋)
6463eqcomd 2183 . . . . . . 7 (𝜑𝑋 = ((𝑋 ∖ {𝑀}) ∪ {𝑀}))
65 f1oeq23 5453 . . . . . . 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 411 . . . . . 6 (𝜑 → ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):(1...(♯‘𝑋))–1-1-onto𝑋 ↔ (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)})–1-1-onto→((𝑋 ∖ {𝑀}) ∪ {𝑀})))
6766adantr 276 . . . . 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 167 . . . 4 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):(1...(♯‘𝑋))–1-1-onto𝑋)
693ad2antrr 488 . . . . . 6 (((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → 𝑋 ∈ Fin)
701ad2antrr 488 . . . . . 6 (((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → 𝑋 ⊆ ℤ)
714ad2antrr 488 . . . . . 6 (((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → 𝑀𝑋)
72 zfz1isolem1.m . . . . . . 7 (𝜑 → ∀𝑧𝑋 𝑧𝑀)
7372ad2antrr 488 . . . . . 6 (((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → ∀𝑧𝑋 𝑧𝑀)
74 simplr 528 . . . . . 6 (((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))
75 simprl 529 . . . . . 6 (((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → 𝑎 ∈ (1...(♯‘𝑋)))
76 simprr 531 . . . . . 6 (((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → 𝑏 ∈ (1...(♯‘𝑋)))
7769, 70, 71, 73, 74, 75, 76zfz1isolemiso 10819 . . . . 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 2559 . . . 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 5226 . . . 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 417 . . 3 ((𝜑𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) Isom < , < ((1...(♯‘𝑋)), 𝑋))
81 vex 2741 . . . . . . 7 𝑔 ∈ V
8281a1i 9 . . . . . 6 (𝜑𝑔 ∈ V)
83 opexg 4229 . . . . . . . 8 (((♯‘𝑋) ∈ ℕ0𝑀𝑋) → ⟨(♯‘𝑋), 𝑀⟩ ∈ V)
8438, 4, 83syl2anc 411 . . . . . . 7 (𝜑 → ⟨(♯‘𝑋), 𝑀⟩ ∈ V)
85 snexg 4185 . . . . . . 7 (⟨(♯‘𝑋), 𝑀⟩ ∈ V → {⟨(♯‘𝑋), 𝑀⟩} ∈ V)
8684, 85syl 14 . . . . . 6 (𝜑 → {⟨(♯‘𝑋), 𝑀⟩} ∈ V)
87 unexg 4444 . . . . . 6 ((𝑔 ∈ V ∧ {⟨(♯‘𝑋), 𝑀⟩} ∈ V) → (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) ∈ V)
8882, 86, 87syl2anc 411 . . . . 5 (𝜑 → (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) ∈ V)
89 isoeq1 5802 . . . . . 6 (𝑓 = (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) → (𝑓 Isom < , < ((1...(♯‘𝑋)), 𝑋) ↔ (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) Isom < , < ((1...(♯‘𝑋)), 𝑋)))
9089spcegv 2826 . . . . 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 276 . . 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 1898 1 (𝜑 → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑋)), 𝑋))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351   = wceq 1353  wex 1492  wcel 2148  wral 2455  Vcvv 2738  cdif 3127  cun 3128  cin 3129  wss 3130  c0 3423  {csn 3593  cop 3596   class class class wbr 4004  suc csuc 4366  ωcom 4590  1-1-ontowf1o 5216  cfv 5217   Isom wiso 5218  (class class class)co 5875  cen 6738  Fincfn 6740  1c1 7812   + caddc 7814   < clt 7992  cle 7993  cmin 8128  0cn0 9176  cz 9253  ...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-addcom 7911  ax-addass 7913  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927
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-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-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-inn 8920  df-n0 9177  df-z 9254  df-uz 9529  df-fz 10009  df-ihash 10756
This theorem is referenced by:  zfz1iso  10821
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