Theorem eldioph2lem1 41987

Description: Lemma for eldioph2 41989. 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 12478	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
2	1	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℝ)
3	2	recnd 11239	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℂ)
4		ax-1cn 11164	. . . . . . . 8 ⊢ 1 ∈ ℂ
5		addcom 11397	. . . . . . . 8 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 + 1) = (1 + 𝑁))
6	3, 4, 5	sylancl 585	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (𝑁 + 1) = (1 + 𝑁))
7		diffi 9175	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ (1...𝑁)) ∈ Fin)
8	7	3ad2ant2 1131	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (𝐴 ∖ (1...𝑁)) ∈ Fin)
9		fzfid 13935	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...𝑁) ∈ Fin)
10		disjdifr 4464	. . . . . . . . . 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 14339	. . . . . . . . 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 1368	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁))) = ((♯‘(𝐴 ∖ (1...𝑁))) + (♯‘(1...𝑁))))
14		uncom 4145	. . . . . . . . . 10 ⊢ ((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁)) = ((1...𝑁) ∪ (𝐴 ∖ (1...𝑁)))
15		simp3 1135	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...𝑁) ⊆ 𝐴)
16		undif 4473	. . . . . . . . . . 11 ⊢ ((1...𝑁) ⊆ 𝐴 ↔ ((1...𝑁) ∪ (𝐴 ∖ (1...𝑁))) = 𝐴)
17	15, 16	sylib 217	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((1...𝑁) ∪ (𝐴 ∖ (1...𝑁))) = 𝐴)
18	14, 17	eqtrid 2776	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁)) = 𝐴)
19	18	fveq2d 6885	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁))) = (♯‘𝐴))
20		hashfz1 14303	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
21	20	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘(1...𝑁)) = 𝑁)
22	21	oveq2d 7417	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((♯‘(𝐴 ∖ (1...𝑁))) + (♯‘(1...𝑁))) = ((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))
23	13, 19, 22	3eqtr3d 2772	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘𝐴) = ((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))
24	6, 23	oveq12d 7419	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝑁 + 1)...(♯‘𝐴)) = ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)))
25	24	fveq2d 6885	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((𝑁 + 1)...(♯‘𝐴))) = (♯‘((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))))
26		1zzd 12590	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 1 ∈ ℤ)
27		hashcl 14313	. . . . . . . . . 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 12581	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘(𝐴 ∖ (1...𝑁))) ∈ ℤ)
30		nn0z 12580	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
31	30	3ad2ant1 1130	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℤ)
32		fzen 13515	. . . . . . . 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 1368	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...(♯‘(𝐴 ∖ (1...𝑁)))) ≈ ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)))
34	33	ensymd 8997	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)) ≈ (1...(♯‘(𝐴 ∖ (1...𝑁)))))
35		fzfi 13934	. . . . . . 7 ⊢ ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)) ∈ Fin
36		fzfi 13934	. . . . . . 7 ⊢ (1...(♯‘(𝐴 ∖ (1...𝑁)))) ∈ Fin
37		hashen 14304	. . . . . . 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 689	. . . . . 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 233	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))) = (♯‘(1...(♯‘(𝐴 ∖ (1...𝑁))))))
40		hashfz1 14303	. . . . . 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 2768	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((𝑁 + 1)...(♯‘𝐴))) = (♯‘(𝐴 ∖ (1...𝑁))))
43		fzfi 13934	. . . . 5 ⊢ ((𝑁 + 1)...(♯‘𝐴)) ∈ Fin
44		hashen 14304	. . . . 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 586	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((♯‘((𝑁 + 1)...(♯‘𝐴))) = (♯‘(𝐴 ∖ (1...𝑁))) ↔ ((𝑁 + 1)...(♯‘𝐴)) ≈ (𝐴 ∖ (1...𝑁))))
46	42, 45	mpbid 231	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝑁 + 1)...(♯‘𝐴)) ≈ (𝐴 ∖ (1...𝑁)))
47		bren 8945	. . 3 ⊢ (((𝑁 + 1)...(♯‘𝐴)) ≈ (𝐴 ∖ (1...𝑁)) ↔ ∃𝑎 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁)))
48	46, 47	sylib 217	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ∃𝑎 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁)))
49		simpl1 1188	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℕ0)
50	49	nn0zd 12581	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℤ)
51		simpl2 1189	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝐴 ∈ Fin)
52		hashcl 14313	. . . . . 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 12581	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (♯‘𝐴) ∈ ℤ)
55		nn0addge2 12516	. . . . . . 7 ⊢ ((𝑁 ∈ ℝ ∧ (♯‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0) → 𝑁 ≤ ((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))
56	2, 28, 55	syl2anc 583	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ≤ ((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))
57	56, 23	breqtrrd 5166	. . . . 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 12825	. . . 4 ⊢ ((♯‘𝐴) ∈ (ℤ≥‘𝑁) ↔ (𝑁 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑁 ≤ (♯‘𝐴)))
60	50, 54, 58, 59	syl3anbrc 1340	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (♯‘𝐴) ∈ (ℤ≥‘𝑁))
61		vex 3470	. . . . 5 ⊢ 𝑎 ∈ V
62		ovex 7434	. . . . . 6 ⊢ (1...𝑁) ∈ V
63		resiexg 7898	. . . . . 6 ⊢ ((1...𝑁) ∈ V → ( I ↾ (1...𝑁)) ∈ V)
64	62, 63	ax-mp 5	. . . . 5 ⊢ ( I ↾ (1...𝑁)) ∈ V
65	61, 64	unex 7726	. . . 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 6861	. . . . . 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 4193	. . . . . 6 ⊢ (((𝑁 + 1)...(♯‘𝐴)) ∩ (1...𝑁)) = ((1...𝑁) ∩ ((𝑁 + 1)...(♯‘𝐴)))
71	49	nn0red 12530	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℝ)
72	71	ltp1d 12141	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 < (𝑁 + 1))
73		fzdisj 13525	. . . . . . 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 2776	. . . . 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 6842	. . . . 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 836	. . . 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 4145	. . . . . 6 ⊢ (((𝑁 + 1)...(♯‘𝐴)) ∪ (1...𝑁)) = ((1...𝑁) ∪ ((𝑁 + 1)...(♯‘𝐴)))
80		fzsplit1nn0 41981	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (♯‘𝐴)) → (1...(♯‘𝐴)) = ((1...𝑁) ∪ ((𝑁 + 1)...(♯‘𝐴))))
81	49, 53, 58, 80	syl3anc 1368	. . . . . 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 2783	. . . . 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 1190	. . . . . . 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 217	. . . . . 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 2776	. . . . 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 6814	. . . . 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 583	. . . 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 231	. . 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 5986	. . . 4 ⊢ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ((𝑎 ↾ (1...𝑁)) ∪ (( I ↾ (1...𝑁)) ↾ (1...𝑁)))
90		dmres 5993	. . . . . . . 8 ⊢ dom (𝑎 ↾ (1...𝑁)) = ((1...𝑁) ∩ dom 𝑎)
91		f1odm 6827	. . . . . . . . . . 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 4204	. . . . . . . . 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 2764	. . . . . . . 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 2776	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → dom (𝑎 ↾ (1...𝑁)) = ∅)
96		relres 6000	. . . . . . . 8 ⊢ Rel (𝑎 ↾ (1...𝑁))
97		reldm0 5917	. . . . . . . 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 233	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (𝑎 ↾ (1...𝑁)) = ∅)
100		residm 6004	. . . . . . 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 4156	. . . . 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 4145	. . . . . 6 ⊢ (∅ ∪ ( I ↾ (1...𝑁))) = (( I ↾ (1...𝑁)) ∪ ∅)
104		un0 4382	. . . . . 6 ⊢ (( I ↾ (1...𝑁)) ∪ ∅) = ( I ↾ (1...𝑁))
105	103, 104	eqtri 2752	. . . . 5 ⊢ (∅ ∪ ( I ↾ (1...𝑁))) = ( I ↾ (1...𝑁))
106	102, 105	eqtrdi 2780	. . . 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 2776	. . 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 7409	. . . . . 6 ⊢ (𝑑 = (♯‘𝐴) → (1...𝑑) = (1...(♯‘𝐴)))
109	108	f1oeq2d 6819	. . . . 5 ⊢ (𝑑 = (♯‘𝐴) → (𝑒:(1...𝑑)–1-1-onto→𝐴 ↔ 𝑒:(1...(♯‘𝐴))–1-1-onto→𝐴))
110	109	anbi1d 629	. . . 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 6811	. . . . 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 5965	. . . . . 6 ⊢ (𝑒 = (𝑎 ∪ ( I ↾ (1...𝑁))) → (𝑒 ↾ (1...𝑁)) = ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)))
113	112	eqeq1d 2726	. . . . 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 630	. . . 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 3616	. . 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 1371	. 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 1930	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 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∃wrex 3062  Vcvv 3466   ∖ cdif 3937   ∪ cun 3938   ∩ cin 3939   ⊆ wss 3940  ∅c0 4314   class class class wbr 5138   I cid 5563  dom cdm 5666   ↾ cres 5668  Rel wrel 5671  –1-1-onto→wf1o 6532  ‘cfv 6533  (class class class)co 7401   ≈ cen 8932  Fincfn 8935  ℂcc 11104  ℝcr 11105  1c1 11107   + caddc 11109   < clt 11245   ≤ cle 11246  ℕ₀cn0 12469  ℤcz 12555  ℤ_≥cuz 12819  ...cfz 13481  ♯chash 14287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-oadd 8465  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-dju 9892  df-card 9930  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-n0 12470  df-z 12556  df-uz 12820  df-fz 13482  df-hash 14288

This theorem is referenced by:  eldioph2  41989