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Theorem zfz1isolem1 10949

Description: Lemma for zfz1iso 10950. Existence of an order isomorphism given the existence of shorter isomorphisms. (Contributed by Jim Kingdon, 7-Sep-2022.)

**Hypotheses**
Ref	Expression
zfz1isolem1.k	⊢ (𝜑 → 𝐾 ∈ ω)
zfz1isolem1.h	⊢ (𝜑 → ∀𝑦(((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑦 ≈ 𝐾) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦)))
zfz1isolem1.xz	⊢ (𝜑 → 𝑋 ⊆ ℤ)
zfz1isolem1.xf	⊢ (𝜑 → 𝑋 ∈ Fin)
zfz1isolem1.xs	⊢ (𝜑 → 𝑋 ≈ suc 𝐾)
zfz1isolem1.mx	⊢ (𝜑 → 𝑀 ∈ 𝑋)
zfz1isolem1.m	⊢ (𝜑 → ∀𝑧 ∈ 𝑋 𝑧 ≤ 𝑀)

**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.

Step	Hyp	Ref	Expression
1		zfz1isolem1.xz	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ ℤ)
2	1	ssdifssd 3302	. . . . 5 ⊢ (𝜑 → (𝑋 ∖ {𝑀}) ⊆ ℤ)
3		zfz1isolem1.xf	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ Fin)
4		zfz1isolem1.mx	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝑋)
5		diffisn 6963	. . . . . 6 ⊢ ((𝑋 ∈ Fin ∧ 𝑀 ∈ 𝑋) → (𝑋 ∖ {𝑀}) ∈ Fin)
6	3, 4, 5	syl2anc 411	. . . . 5 ⊢ (𝜑 → (𝑋 ∖ {𝑀}) ∈ Fin)
7		zfz1isolem1.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ω)
8		zfz1isolem1.xs	. . . . . 6 ⊢ (𝜑 → 𝑋 ≈ suc 𝐾)
9		dif1en 6949	. . . . . 6 ⊢ ((𝐾 ∈ ω ∧ 𝑋 ≈ suc 𝐾 ∧ 𝑀 ∈ 𝑋) → (𝑋 ∖ {𝑀}) ≈ 𝐾)
10	7, 8, 4, 9	syl3anc 1249	. . . . 5 ⊢ (𝜑 → (𝑋 ∖ {𝑀}) ≈ 𝐾)
11	2, 6, 10	jca31 309	. . . 4 ⊢ (𝜑 → (((𝑋 ∖ {𝑀}) ⊆ ℤ ∧ (𝑋 ∖ {𝑀}) ∈ Fin) ∧ (𝑋 ∖ {𝑀}) ≈ 𝐾))
12		zfz1isolem1.h	. . . . 5 ⊢ (𝜑 → ∀𝑦(((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑦 ≈ 𝐾) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦)))
13		sseq1 3207	. . . . . . . . . 10 ⊢ (𝑦 = (𝑋 ∖ {𝑀}) → (𝑦 ⊆ ℤ ↔ (𝑋 ∖ {𝑀}) ⊆ ℤ))
14		eleq1 2259	. . . . . . . . . 10 ⊢ (𝑦 = (𝑋 ∖ {𝑀}) → (𝑦 ∈ Fin ↔ (𝑋 ∖ {𝑀}) ∈ Fin))
15	13, 14	anbi12d 473	. . . . . . . . 9 ⊢ (𝑦 = (𝑋 ∖ {𝑀}) → ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ↔ ((𝑋 ∖ {𝑀}) ⊆ ℤ ∧ (𝑋 ∖ {𝑀}) ∈ Fin)))
16		breq1 4037	. . . . . . . . 9 ⊢ (𝑦 = (𝑋 ∖ {𝑀}) → (𝑦 ≈ 𝐾 ↔ (𝑋 ∖ {𝑀}) ≈ 𝐾))
17	15, 16	anbi12d 473	. . . . . . . 8 ⊢ (𝑦 = (𝑋 ∖ {𝑀}) → (((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑦 ≈ 𝐾) ↔ (((𝑋 ∖ {𝑀}) ⊆ ℤ ∧ (𝑋 ∖ {𝑀}) ∈ Fin) ∧ (𝑋 ∖ {𝑀}) ≈ 𝐾)))
18		fveq2 5561	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑋 ∖ {𝑀}) → (♯‘𝑦) = (♯‘(𝑋 ∖ {𝑀})))
19	18	oveq2d 5941	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑋 ∖ {𝑀}) → (1...(♯‘𝑦)) = (1...(♯‘(𝑋 ∖ {𝑀}))))
20		isoeq4 5854	. . . . . . . . . . 11 ⊢ ((1...(♯‘𝑦)) = (1...(♯‘(𝑋 ∖ {𝑀}))) → (𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦) ↔ 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), 𝑦)))
21	19, 20	syl 14	. . . . . . . . . 10 ⊢ (𝑦 = (𝑋 ∖ {𝑀}) → (𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦) ↔ 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), 𝑦)))
22		isoeq5 5855	. . . . . . . . . 10 ⊢ (𝑦 = (𝑋 ∖ {𝑀}) → (𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), 𝑦) ↔ 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))))
23	21, 22	bitrd 188	. . . . . . . . 9 ⊢ (𝑦 = (𝑋 ∖ {𝑀}) → (𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦) ↔ 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))))
24	23	exbidv 1839	. . . . . . . 8 ⊢ (𝑦 = (𝑋 ∖ {𝑀}) → (∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦) ↔ ∃𝑓 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))))
25	17, 24	imbi12d 234	. . . . . . 7 ⊢ (𝑦 = (𝑋 ∖ {𝑀}) → ((((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑦 ≈ 𝐾) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦)) ↔ ((((𝑋 ∖ {𝑀}) ⊆ ℤ ∧ (𝑋 ∖ {𝑀}) ∈ Fin) ∧ (𝑋 ∖ {𝑀}) ≈ 𝐾) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))))
26	25	spcgv 2851	. . . . . 6 ⊢ ((𝑋 ∖ {𝑀}) ∈ Fin → (∀𝑦(((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑦 ≈ 𝐾) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦)) → ((((𝑋 ∖ {𝑀}) ⊆ ℤ ∧ (𝑋 ∖ {𝑀}) ∈ Fin) ∧ (𝑋 ∖ {𝑀}) ≈ 𝐾) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))))
27	6, 26	syl 14	. . . . 5 ⊢ (𝜑 → (∀𝑦(((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑦 ≈ 𝐾) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑦)), 𝑦)) → ((((𝑋 ∖ {𝑀}) ⊆ ℤ ∧ (𝑋 ∖ {𝑀}) ∈ Fin) ∧ (𝑋 ∖ {𝑀}) ≈ 𝐾) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))))
28	12, 27	mpd 13	. . . 4 ⊢ (𝜑 → ((((𝑋 ∖ {𝑀}) ⊆ ℤ ∧ (𝑋 ∖ {𝑀}) ∈ Fin) ∧ (𝑋 ∖ {𝑀}) ≈ 𝐾) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))))
29	11, 28	mpd 13	. . 3 ⊢ (𝜑 → ∃𝑓 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))
30		isoeq1 5851	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})) ↔ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))))
31	30	cbvexv 1933	. . 3 ⊢ (∃𝑓 𝑓 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})) ↔ ∃𝑔 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))
32	29, 31	sylib 122	. 2 ⊢ (𝜑 → ∃𝑔 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))
33		df-isom 5268	. . . . . . . . 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 ↔ (𝑔‘𝑢) < (𝑔‘𝑣))))
34	33	biimpi 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 ↔ (𝑔‘𝑢) < (𝑔‘𝑣))))
35	34	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ 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 ↔ (𝑔‘𝑢) < (𝑔‘𝑣))))
36	35	simpld 112	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → 𝑔:(1...(♯‘(𝑋 ∖ {𝑀})))–1-1-onto→(𝑋 ∖ {𝑀}))
37		hashcl 10890	. . . . . . . . 9 ⊢ (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ₀)
38	3, 37	syl 14	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ₀)
39	38	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → (♯‘𝑋) ∈ ℕ₀)
40	4	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → 𝑀 ∈ 𝑋)
41		f1osng 5548	. . . . . . 7 ⊢ (((♯‘𝑋) ∈ ℕ₀ ∧ 𝑀 ∈ 𝑋) → {⟨(♯‘𝑋), 𝑀⟩}:{(♯‘𝑋)}–1-1-onto→{𝑀})
42	39, 40, 41	syl2anc 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → {⟨(♯‘𝑋), 𝑀⟩}:{(♯‘𝑋)}–1-1-onto→{𝑀})
43		hashdifsn 10928	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ Fin ∧ 𝑀 ∈ 𝑋) → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1))
44	3, 4, 43	syl2anc 411	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝑋 ∖ {𝑀})) = ((♯‘𝑋) − 1))
45	44	oveq1d 5940	. . . . . . . . . . 11 ⊢ (𝜑 → ((♯‘(𝑋 ∖ {𝑀})) + 1) = (((♯‘𝑋) − 1) + 1))
46	38	nn0cnd 9321	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑋) ∈ ℂ)
47		1cnd 8059	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℂ)
48	46, 47	npcand 8358	. . . . . . . . . . 11 ⊢ (𝜑 → (((♯‘𝑋) − 1) + 1) = (♯‘𝑋))
49	45, 48	eqtrd 2229	. . . . . . . . . 10 ⊢ (𝜑 → ((♯‘(𝑋 ∖ {𝑀})) + 1) = (♯‘𝑋))
50	49	sneqd 3636	. . . . . . . . 9 ⊢ (𝜑 → {((♯‘(𝑋 ∖ {𝑀})) + 1)} = {(♯‘𝑋)})
51	50	ineq2d 3365	. . . . . . . 8 ⊢ (𝜑 → ((1...(♯‘(𝑋 ∖ {𝑀}))) ∩ {((♯‘(𝑋 ∖ {𝑀})) + 1)}) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∩ {(♯‘𝑋)}))
52		fzp1disj 10172	. . . . . . . 8 ⊢ ((1...(♯‘(𝑋 ∖ {𝑀}))) ∩ {((♯‘(𝑋 ∖ {𝑀})) + 1)}) = ∅
53	51, 52	eqtr3di 2244	. . . . . . 7 ⊢ (𝜑 → ((1...(♯‘(𝑋 ∖ {𝑀}))) ∩ {(♯‘𝑋)}) = ∅)
54	53	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → ((1...(♯‘(𝑋 ∖ {𝑀}))) ∩ {(♯‘𝑋)}) = ∅)
55		incom 3356	. . . . . . . 8 ⊢ ((𝑋 ∖ {𝑀}) ∩ {𝑀}) = ({𝑀} ∩ (𝑋 ∖ {𝑀}))
56		disjdif 3524	. . . . . . . 8 ⊢ ({𝑀} ∩ (𝑋 ∖ {𝑀})) = ∅
57	55, 56	eqtri 2217	. . . . . . 7 ⊢ ((𝑋 ∖ {𝑀}) ∩ {𝑀}) = ∅
58	57	a1i 9	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → ((𝑋 ∖ {𝑀}) ∩ {𝑀}) = ∅)
59		f1oun 5527	. . . . . 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}))
60	36, 42, 54, 58, 59	syl22anc 1250	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)})–1-1-onto→((𝑋 ∖ {𝑀}) ∪ {𝑀}))
61	3, 4	zfz1isolemsplit 10947	. . . . . . 7 ⊢ (𝜑 → (1...(♯‘𝑋)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)}))
62		fidifsnid 6941	. . . . . . . . 9 ⊢ ((𝑋 ∈ Fin ∧ 𝑀 ∈ 𝑋) → ((𝑋 ∖ {𝑀}) ∪ {𝑀}) = 𝑋)
63	3, 4, 62	syl2anc 411	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 ∖ {𝑀}) ∪ {𝑀}) = 𝑋)
64	63	eqcomd 2202	. . . . . . 7 ⊢ (𝜑 → 𝑋 = ((𝑋 ∖ {𝑀}) ∪ {𝑀}))
65		f1oeq23 5498	. . . . . . 7 ⊢ (((1...(♯‘𝑋)) = ((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)}) ∧ 𝑋 = ((𝑋 ∖ {𝑀}) ∪ {𝑀})) → ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):(1...(♯‘𝑋))–1-1-onto→𝑋 ↔ (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)})–1-1-onto→((𝑋 ∖ {𝑀}) ∪ {𝑀})))
66	61, 64, 65	syl2anc 411	. . . . . 6 ⊢ (𝜑 → ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):(1...(♯‘𝑋))–1-1-onto→𝑋 ↔ (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)})–1-1-onto→((𝑋 ∖ {𝑀}) ∪ {𝑀})))
67	66	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):(1...(♯‘𝑋))–1-1-onto→𝑋 ↔ (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):((1...(♯‘(𝑋 ∖ {𝑀}))) ∪ {(♯‘𝑋)})–1-1-onto→((𝑋 ∖ {𝑀}) ∪ {𝑀})))
68	60, 67	mpbird 167	. . . 4 ⊢ ((𝜑 ∧ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):(1...(♯‘𝑋))–1-1-onto→𝑋)
69	3	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → 𝑋 ∈ Fin)
70	1	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → 𝑋 ⊆ ℤ)
71	4	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → 𝑀 ∈ 𝑋)
72		zfz1isolem1.m	. . . . . . 7 ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 𝑧 ≤ 𝑀)
73	72	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → ∀𝑧 ∈ 𝑋 𝑧 ≤ 𝑀)
74		simplr 528	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀})))
75		simprl 529	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → 𝑎 ∈ (1...(♯‘𝑋)))
76		simprr 531	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → 𝑏 ∈ (1...(♯‘𝑋)))
77	69, 70, 71, 73, 74, 75, 76	zfz1isolemiso 10948	. . . . 5 ⊢ (((𝜑 ∧ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) ∧ (𝑎 ∈ (1...(♯‘𝑋)) ∧ 𝑏 ∈ (1...(♯‘𝑋)))) → (𝑎 < 𝑏 ↔ ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝑎) < ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝑏)))
78	77	ralrimivva 2579	. . . 4 ⊢ ((𝜑 ∧ 𝑔 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 5268	. . . 4 ⊢ ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) Isom < , < ((1...(♯‘𝑋)), 𝑋) ↔ ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}):(1...(♯‘𝑋))–1-1-onto→𝑋 ∧ ∀𝑎 ∈ (1...(♯‘𝑋))∀𝑏 ∈ (1...(♯‘𝑋))(𝑎 < 𝑏 ↔ ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝑎) < ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩})‘𝑏))))
80	68, 78, 79	sylanbrc 417	. . 3 ⊢ ((𝜑 ∧ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) Isom < , < ((1...(♯‘𝑋)), 𝑋))
81		vex 2766	. . . . . . 7 ⊢ 𝑔 ∈ V
82	81	a1i 9	. . . . . 6 ⊢ (𝜑 → 𝑔 ∈ V)
83		opexg 4262	. . . . . . . 8 ⊢ (((♯‘𝑋) ∈ ℕ₀ ∧ 𝑀 ∈ 𝑋) → ⟨(♯‘𝑋), 𝑀⟩ ∈ V)
84	38, 4, 83	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → ⟨(♯‘𝑋), 𝑀⟩ ∈ V)
85		snexg 4218	. . . . . . 7 ⊢ (⟨(♯‘𝑋), 𝑀⟩ ∈ V → {⟨(♯‘𝑋), 𝑀⟩} ∈ V)
86	84, 85	syl 14	. . . . . 6 ⊢ (𝜑 → {⟨(♯‘𝑋), 𝑀⟩} ∈ V)
87		unexg 4479	. . . . . 6 ⊢ ((𝑔 ∈ V ∧ {⟨(♯‘𝑋), 𝑀⟩} ∈ V) → (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) ∈ V)
88	82, 86, 87	syl2anc 411	. . . . 5 ⊢ (𝜑 → (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) ∈ V)
89		isoeq1 5851	. . . . . 6 ⊢ (𝑓 = (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) → (𝑓 Isom < , < ((1...(♯‘𝑋)), 𝑋) ↔ (𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) Isom < , < ((1...(♯‘𝑋)), 𝑋)))
90	89	spcegv 2852	. . . . 5 ⊢ ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) ∈ V → ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) Isom < , < ((1...(♯‘𝑋)), 𝑋) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑋)), 𝑋)))
91	88, 90	syl 14	. . . 4 ⊢ (𝜑 → ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) Isom < , < ((1...(♯‘𝑋)), 𝑋) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑋)), 𝑋)))
92	91	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → ((𝑔 ∪ {⟨(♯‘𝑋), 𝑀⟩}) Isom < , < ((1...(♯‘𝑋)), 𝑋) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑋)), 𝑋)))
93	80, 92	mpd 13	. 2 ⊢ ((𝜑 ∧ 𝑔 Isom < , < ((1...(♯‘(𝑋 ∖ {𝑀}))), (𝑋 ∖ {𝑀}))) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑋)), 𝑋))
94	32, 93	exlimddv 1913	1 ⊢ (𝜑 → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑋)), 𝑋))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∀wal 1362 = wceq 1364 ∃wex 1506 ∈ wcel 2167 ∀wral 2475 Vcvv 2763 ∖ cdif 3154 ∪ cun 3155 ∩ cin 3156 ⊆ wss 3157 ∅c0 3451 {csn 3623 ⟨cop 3626 class class class wbr 4034 suc csuc 4401 ωcom 4627 –1-1-onto→wf1o 5258 ‘cfv 5259 Isom wiso 5260 (class class class)co 5925 ≈ cen 6806 Fincfn 6808 1c1 7897 + caddc 7899 < clt 8078 ≤ cle 8079 − cmin 8214 ℕ₀cn0 9266 ℤcz 9343 ...cfz 10100 ♯chash 10884

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-addcom 7996 ax-addass 7998 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-0id 8004 ax-rnegex 8005 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-isom 5268 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-irdg 6437 df-frec 6458 df-1o 6483 df-oadd 6487 df-er 6601 df-en 6809 df-dom 6810 df-fin 6811 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-inn 9008 df-n0 9267 df-z 9344 df-uz 9619 df-fz 10101 df-ihash 10885

This theorem is referenced by: zfz1iso 10950

W3C validator