Theorem eldioph2lem1 43146

Description: Lemma for eldioph2 43148. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Assertion

Ref	Expression
eldioph2lem1	⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ∃𝑑 ∈ (ℤ≥‘𝑁)∃𝑒 ∈ V (𝑒:(1...𝑑)–1-1-onto→𝐴 ∧ (𝑒 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))

Distinct variable groups:   𝐴,𝑑,𝑒   𝑁,𝑑,𝑒

Proof of Theorem eldioph2lem1

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0re 12424	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
2	1	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℝ)
3	2	recnd 11174	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℂ)
4		ax-1cn 11098	. . . . . . . 8 ⊢ 1 ∈ ℂ
5		addcom 11333	. . . . . . . 8 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 + 1) = (1 + 𝑁))
6	3, 4, 5	sylancl 587	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (𝑁 + 1) = (1 + 𝑁))
7		diffi 9113	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ (1...𝑁)) ∈ Fin)
8	7	3ad2ant2 1135	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (𝐴 ∖ (1...𝑁)) ∈ Fin)
9		fzfid 13910	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...𝑁) ∈ Fin)
10		disjdifr 4427	. . . . . . . . . 10 ⊢ ((𝐴 ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅
11	10	a1i 11	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝐴 ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅)
12		hashun 14319	. . . . . . . . 9 ⊢ (((𝐴 ∖ (1...𝑁)) ∈ Fin ∧ (1...𝑁) ∈ Fin ∧ ((𝐴 ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅) → (♯‘((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁))) = ((♯‘(𝐴 ∖ (1...𝑁))) + (♯‘(1...𝑁))))
13	8, 9, 11, 12	syl3anc 1374	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁))) = ((♯‘(𝐴 ∖ (1...𝑁))) + (♯‘(1...𝑁))))
14		uncom 4112	. . . . . . . . . 10 ⊢ ((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁)) = ((1...𝑁) ∪ (𝐴 ∖ (1...𝑁)))
15		simp3 1139	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...𝑁) ⊆ 𝐴)
16		undif 4436	. . . . . . . . . . 11 ⊢ ((1...𝑁) ⊆ 𝐴 ↔ ((1...𝑁) ∪ (𝐴 ∖ (1...𝑁))) = 𝐴)
17	15, 16	sylib 218	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((1...𝑁) ∪ (𝐴 ∖ (1...𝑁))) = 𝐴)
18	14, 17	eqtrid 2784	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁)) = 𝐴)
19	18	fveq2d 6848	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁))) = (♯‘𝐴))
20		hashfz1 14283	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
21	20	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘(1...𝑁)) = 𝑁)
22	21	oveq2d 7386	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((♯‘(𝐴 ∖ (1...𝑁))) + (♯‘(1...𝑁))) = ((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))
23	13, 19, 22	3eqtr3d 2780	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘𝐴) = ((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))
24	6, 23	oveq12d 7388	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝑁 + 1)...(♯‘𝐴)) = ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)))
25	24	fveq2d 6848	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((𝑁 + 1)...(♯‘𝐴))) = (♯‘((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))))
26		1zzd 12536	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 1 ∈ ℤ)
27		hashcl 14293	. . . . . . . . . 10 ⊢ ((𝐴 ∖ (1...𝑁)) ∈ Fin → (♯‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0)
28	8, 27	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0)
29	28	nn0zd 12527	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘(𝐴 ∖ (1...𝑁))) ∈ ℤ)
30		nn0z 12526	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
31	30	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℤ)
32		fzen 13471	. . . . . . . 8 ⊢ ((1 ∈ ℤ ∧ (♯‘(𝐴 ∖ (1...𝑁))) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (1...(♯‘(𝐴 ∖ (1...𝑁)))) ≈ ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)))
33	26, 29, 31, 32	syl3anc 1374	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...(♯‘(𝐴 ∖ (1...𝑁)))) ≈ ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)))
34	33	ensymd 8956	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)) ≈ (1...(♯‘(𝐴 ∖ (1...𝑁)))))
35		fzfi 13909	. . . . . . 7 ⊢ ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)) ∈ Fin
36		fzfi 13909	. . . . . . 7 ⊢ (1...(♯‘(𝐴 ∖ (1...𝑁)))) ∈ Fin
37		hashen 14284	. . . . . . 7 ⊢ ((((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)) ∈ Fin ∧ (1...(♯‘(𝐴 ∖ (1...𝑁)))) ∈ Fin) → ((♯‘((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))) = (♯‘(1...(♯‘(𝐴 ∖ (1...𝑁))))) ↔ ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)) ≈ (1...(♯‘(𝐴 ∖ (1...𝑁))))))
38	35, 36, 37	mp2an 693	. . . . . 6 ⊢ ((♯‘((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))) = (♯‘(1...(♯‘(𝐴 ∖ (1...𝑁))))) ↔ ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)) ≈ (1...(♯‘(𝐴 ∖ (1...𝑁)))))
39	34, 38	sylibr 234	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))) = (♯‘(1...(♯‘(𝐴 ∖ (1...𝑁))))))
40		hashfz1 14283	. . . . . 6 ⊢ ((♯‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0 → (♯‘(1...(♯‘(𝐴 ∖ (1...𝑁))))) = (♯‘(𝐴 ∖ (1...𝑁))))
41	28, 40	syl 17	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘(1...(♯‘(𝐴 ∖ (1...𝑁))))) = (♯‘(𝐴 ∖ (1...𝑁))))
42	25, 39, 41	3eqtrd 2776	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((𝑁 + 1)...(♯‘𝐴))) = (♯‘(𝐴 ∖ (1...𝑁))))
43		fzfi 13909	. . . . 5 ⊢ ((𝑁 + 1)...(♯‘𝐴)) ∈ Fin
44		hashen 14284	. . . . 5 ⊢ ((((𝑁 + 1)...(♯‘𝐴)) ∈ Fin ∧ (𝐴 ∖ (1...𝑁)) ∈ Fin) → ((♯‘((𝑁 + 1)...(♯‘𝐴))) = (♯‘(𝐴 ∖ (1...𝑁))) ↔ ((𝑁 + 1)...(♯‘𝐴)) ≈ (𝐴 ∖ (1...𝑁))))
45	43, 8, 44	sylancr 588	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((♯‘((𝑁 + 1)...(♯‘𝐴))) = (♯‘(𝐴 ∖ (1...𝑁))) ↔ ((𝑁 + 1)...(♯‘𝐴)) ≈ (𝐴 ∖ (1...𝑁))))
46	42, 45	mpbid 232	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝑁 + 1)...(♯‘𝐴)) ≈ (𝐴 ∖ (1...𝑁)))
47		bren 8907	. . 3 ⊢ (((𝑁 + 1)...(♯‘𝐴)) ≈ (𝐴 ∖ (1...𝑁)) ↔ ∃𝑎 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁)))
48	46, 47	sylib 218	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ∃𝑎 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁)))
49		simpl1 1193	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℕ0)
50	49	nn0zd 12527	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℤ)
51		simpl2 1194	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝐴 ∈ Fin)
52		hashcl 14293	. . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
53	51, 52	syl 17	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (♯‘𝐴) ∈ ℕ0)
54	53	nn0zd 12527	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (♯‘𝐴) ∈ ℤ)
55		nn0addge2 12462	. . . . . . 7 ⊢ ((𝑁 ∈ ℝ ∧ (♯‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0) → 𝑁 ≤ ((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))
56	2, 28, 55	syl2anc 585	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ≤ ((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))
57	56, 23	breqtrrd 5128	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ≤ (♯‘𝐴))
58	57	adantr 480	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ≤ (♯‘𝐴))
59		eluz2 12771	. . . 4 ⊢ ((♯‘𝐴) ∈ (ℤ≥‘𝑁) ↔ (𝑁 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑁 ≤ (♯‘𝐴)))
60	50, 54, 58, 59	syl3anbrc 1345	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (♯‘𝐴) ∈ (ℤ≥‘𝑁))
61		vex 3446	. . . . 5 ⊢ 𝑎 ∈ V
62		ovex 7403	. . . . . 6 ⊢ (1...𝑁) ∈ V
63		resiexg 7866	. . . . . 6 ⊢ ((1...𝑁) ∈ V → ( I ↾ (1...𝑁)) ∈ V)
64	62, 63	ax-mp 5	. . . . 5 ⊢ ( I ↾ (1...𝑁)) ∈ V
65	61, 64	unex 7701	. . . 4 ⊢ (𝑎 ∪ ( I ↾ (1...𝑁))) ∈ V
66	65	a1i 11	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (𝑎 ∪ ( I ↾ (1...𝑁))) ∈ V)
67		simpr 484	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁)))
68		f1oi 6822	. . . . . 6 ⊢ ( I ↾ (1...𝑁)):(1...𝑁)–1-1-onto→(1...𝑁)
69	68	a1i 11	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ( I ↾ (1...𝑁)):(1...𝑁)–1-1-onto→(1...𝑁))
70		incom 4163	. . . . . 6 ⊢ (((𝑁 + 1)...(♯‘𝐴)) ∩ (1...𝑁)) = ((1...𝑁) ∩ ((𝑁 + 1)...(♯‘𝐴)))
71	49	nn0red 12477	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℝ)
72	71	ltp1d 12086	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 < (𝑁 + 1))
73		fzdisj 13481	. . . . . . 7 ⊢ (𝑁 < (𝑁 + 1) → ((1...𝑁) ∩ ((𝑁 + 1)...(♯‘𝐴))) = ∅)
74	72, 73	syl 17	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ((1...𝑁) ∩ ((𝑁 + 1)...(♯‘𝐴))) = ∅)
75	70, 74	eqtrid 2784	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (((𝑁 + 1)...(♯‘𝐴)) ∩ (1...𝑁)) = ∅)
76	10	a1i 11	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ((𝐴 ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅)
77		f1oun 6803	. . . . 5 ⊢ (((𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁)) ∧ ( I ↾ (1...𝑁)):(1...𝑁)–1-1-onto→(1...𝑁)) ∧ ((((𝑁 + 1)...(♯‘𝐴)) ∩ (1...𝑁)) = ∅ ∧ ((𝐴 ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅)) → (𝑎 ∪ ( I ↾ (1...𝑁))):(((𝑁 + 1)...(♯‘𝐴)) ∪ (1...𝑁))–1-1-onto→((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁)))
78	67, 69, 75, 76, 77	syl22anc 839	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (𝑎 ∪ ( I ↾ (1...𝑁))):(((𝑁 + 1)...(♯‘𝐴)) ∪ (1...𝑁))–1-1-onto→((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁)))
79		uncom 4112	. . . . . 6 ⊢ (((𝑁 + 1)...(♯‘𝐴)) ∪ (1...𝑁)) = ((1...𝑁) ∪ ((𝑁 + 1)...(♯‘𝐴)))
80		fzsplit1nn0 43140	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (♯‘𝐴)) → (1...(♯‘𝐴)) = ((1...𝑁) ∪ ((𝑁 + 1)...(♯‘𝐴))))
81	49, 53, 58, 80	syl3anc 1374	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (1...(♯‘𝐴)) = ((1...𝑁) ∪ ((𝑁 + 1)...(♯‘𝐴))))
82	79, 81	eqtr4id 2791	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (((𝑁 + 1)...(♯‘𝐴)) ∪ (1...𝑁)) = (1...(♯‘𝐴)))
83		simpl3 1195	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (1...𝑁) ⊆ 𝐴)
84	83, 16	sylib 218	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ((1...𝑁) ∪ (𝐴 ∖ (1...𝑁))) = 𝐴)
85	14, 84	eqtrid 2784	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁)) = 𝐴)
86		f1oeq23 6775	. . . . 5 ⊢ (((((𝑁 + 1)...(♯‘𝐴)) ∪ (1...𝑁)) = (1...(♯‘𝐴)) ∧ ((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁)) = 𝐴) → ((𝑎 ∪ ( I ↾ (1...𝑁))):(((𝑁 + 1)...(♯‘𝐴)) ∪ (1...𝑁))–1-1-onto→((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁)) ↔ (𝑎 ∪ ( I ↾ (1...𝑁))):(1...(♯‘𝐴))–1-1-onto→𝐴))
87	82, 85, 86	syl2anc 585	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ((𝑎 ∪ ( I ↾ (1...𝑁))):(((𝑁 + 1)...(♯‘𝐴)) ∪ (1...𝑁))–1-1-onto→((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁)) ↔ (𝑎 ∪ ( I ↾ (1...𝑁))):(1...(♯‘𝐴))–1-1-onto→𝐴))
88	78, 87	mpbid 232	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (𝑎 ∪ ( I ↾ (1...𝑁))):(1...(♯‘𝐴))–1-1-onto→𝐴)
89		resundir 5963	. . . 4 ⊢ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ((𝑎 ↾ (1...𝑁)) ∪ (( I ↾ (1...𝑁)) ↾ (1...𝑁)))
90		dmres 5981	. . . . . . . 8 ⊢ dom (𝑎 ↾ (1...𝑁)) = ((1...𝑁) ∩ dom 𝑎)
91		f1odm 6788	. . . . . . . . . . 11 ⊢ (𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁)) → dom 𝑎 = ((𝑁 + 1)...(♯‘𝐴)))
92	91	adantl 481	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → dom 𝑎 = ((𝑁 + 1)...(♯‘𝐴)))
93	92	ineq2d 4174	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ((1...𝑁) ∩ dom 𝑎) = ((1...𝑁) ∩ ((𝑁 + 1)...(♯‘𝐴))))
94	93, 74	eqtrd 2772	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ((1...𝑁) ∩ dom 𝑎) = ∅)
95	90, 94	eqtrid 2784	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → dom (𝑎 ↾ (1...𝑁)) = ∅)
96		relres 5974	. . . . . . . 8 ⊢ Rel (𝑎 ↾ (1...𝑁))
97		reldm0 5887	. . . . . . . 8 ⊢ (Rel (𝑎 ↾ (1...𝑁)) → ((𝑎 ↾ (1...𝑁)) = ∅ ↔ dom (𝑎 ↾ (1...𝑁)) = ∅))
98	96, 97	ax-mp 5	. . . . . . 7 ⊢ ((𝑎 ↾ (1...𝑁)) = ∅ ↔ dom (𝑎 ↾ (1...𝑁)) = ∅)
99	95, 98	sylibr 234	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (𝑎 ↾ (1...𝑁)) = ∅)
100		residm 5979	. . . . . . 7 ⊢ (( I ↾ (1...𝑁)) ↾ (1...𝑁)) = ( I ↾ (1...𝑁))
101	100	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (( I ↾ (1...𝑁)) ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))
102	99, 101	uneq12d 4123	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ((𝑎 ↾ (1...𝑁)) ∪ (( I ↾ (1...𝑁)) ↾ (1...𝑁))) = (∅ ∪ ( I ↾ (1...𝑁))))
103		uncom 4112	. . . . . 6 ⊢ (∅ ∪ ( I ↾ (1...𝑁))) = (( I ↾ (1...𝑁)) ∪ ∅)
104		un0 4348	. . . . . 6 ⊢ (( I ↾ (1...𝑁)) ∪ ∅) = ( I ↾ (1...𝑁))
105	103, 104	eqtri 2760	. . . . 5 ⊢ (∅ ∪ ( I ↾ (1...𝑁))) = ( I ↾ (1...𝑁))
106	102, 105	eqtrdi 2788	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ((𝑎 ↾ (1...𝑁)) ∪ (( I ↾ (1...𝑁)) ↾ (1...𝑁))) = ( I ↾ (1...𝑁)))
107	89, 106	eqtrid 2784	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))
108		oveq2 7378	. . . . . 6 ⊢ (𝑑 = (♯‘𝐴) → (1...𝑑) = (1...(♯‘𝐴)))
109	108	f1oeq2d 6780	. . . . 5 ⊢ (𝑑 = (♯‘𝐴) → (𝑒:(1...𝑑)–1-1-onto→𝐴 ↔ 𝑒:(1...(♯‘𝐴))–1-1-onto→𝐴))
110	109	anbi1d 632	. . . 4 ⊢ (𝑑 = (♯‘𝐴) → ((𝑒:(1...𝑑)–1-1-onto→𝐴 ∧ (𝑒 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) ↔ (𝑒:(1...(♯‘𝐴))–1-1-onto→𝐴 ∧ (𝑒 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))))
111		f1oeq1 6772	. . . . 5 ⊢ (𝑒 = (𝑎 ∪ ( I ↾ (1...𝑁))) → (𝑒:(1...(♯‘𝐴))–1-1-onto→𝐴 ↔ (𝑎 ∪ ( I ↾ (1...𝑁))):(1...(♯‘𝐴))–1-1-onto→𝐴))
112		reseq1 5942	. . . . . 6 ⊢ (𝑒 = (𝑎 ∪ ( I ↾ (1...𝑁))) → (𝑒 ↾ (1...𝑁)) = ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)))
113	112	eqeq1d 2739	. . . . 5 ⊢ (𝑒 = (𝑎 ∪ ( I ↾ (1...𝑁))) → ((𝑒 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)) ↔ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
114	111, 113	anbi12d 633	. . . 4 ⊢ (𝑒 = (𝑎 ∪ ( I ↾ (1...𝑁))) → ((𝑒:(1...(♯‘𝐴))–1-1-onto→𝐴 ∧ (𝑒 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) ↔ ((𝑎 ∪ ( I ↾ (1...𝑁))):(1...(♯‘𝐴))–1-1-onto→𝐴 ∧ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))))
115	110, 114	rspc2ev 3591	. . 3 ⊢ (((♯‘𝐴) ∈ (ℤ≥‘𝑁) ∧ (𝑎 ∪ ( I ↾ (1...𝑁))) ∈ V ∧ ((𝑎 ∪ ( I ↾ (1...𝑁))):(1...(♯‘𝐴))–1-1-onto→𝐴 ∧ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → ∃𝑑 ∈ (ℤ≥‘𝑁)∃𝑒 ∈ V (𝑒:(1...𝑑)–1-1-onto→𝐴 ∧ (𝑒 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
116	60, 66, 88, 107, 115	syl112anc 1377	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ∃𝑑 ∈ (ℤ≥‘𝑁)∃𝑒 ∈ V (𝑒:(1...𝑑)–1-1-onto→𝐴 ∧ (𝑒 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
117	48, 116	exlimddv 1937	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ∃𝑑 ∈ (ℤ≥‘𝑁)∃𝑒 ∈ V (𝑒:(1...𝑑)–1-1-onto→𝐴 ∧ (𝑒 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3062  Vcvv 3442   ∖ cdif 3900   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287   class class class wbr 5100   I cid 5528  dom cdm 5634   ↾ cres 5636  Rel wrel 5639  –1-1-onto→wf1o 6501  ‘cfv 6502  (class class class)co 7370   ≈ cen 8894  Fincfn 8897  ℂcc 11038  ℝcr 11039  1c1 11041   + caddc 11043   < clt 11180   ≤ cle 11181  ℕ₀cn0 12415  ℤcz 12502  ℤ_≥cuz 12765  ...cfz 13437  ♯chash 14267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-oadd 8413  df-er 8647  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-dju 9827  df-card 9865  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-nn 12160  df-n0 12416  df-z 12503  df-uz 12766  df-fz 13438  df-hash 14268

This theorem is referenced by:  eldioph2  43148