Theorem eldioph2lem1 42771

Description: Lemma for eldioph2 42773. 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 12535	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
2	1	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℝ)
3	2	recnd 11289	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℂ)
4		ax-1cn 11213	. . . . . . . 8 ⊢ 1 ∈ ℂ
5		addcom 11447	. . . . . . . 8 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 + 1) = (1 + 𝑁))
6	3, 4, 5	sylancl 586	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (𝑁 + 1) = (1 + 𝑁))
7		diffi 9215	. . . . . . . . . 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 14014	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...𝑁) ∈ Fin)
10		disjdifr 4473	. . . . . . . . . 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 14421	. . . . . . . . 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 1373	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁))) = ((♯‘(𝐴 ∖ (1...𝑁))) + (♯‘(1...𝑁))))
14		uncom 4158	. . . . . . . . . 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 4482	. . . . . . . . . . 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 2789	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁)) = 𝐴)
19	18	fveq2d 6910	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁))) = (♯‘𝐴))
20		hashfz1 14385	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
21	20	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘(1...𝑁)) = 𝑁)
22	21	oveq2d 7447	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((♯‘(𝐴 ∖ (1...𝑁))) + (♯‘(1...𝑁))) = ((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))
23	13, 19, 22	3eqtr3d 2785	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘𝐴) = ((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))
24	6, 23	oveq12d 7449	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝑁 + 1)...(♯‘𝐴)) = ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)))
25	24	fveq2d 6910	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((𝑁 + 1)...(♯‘𝐴))) = (♯‘((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))))
26		1zzd 12648	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 1 ∈ ℤ)
27		hashcl 14395	. . . . . . . . . 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 12639	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘(𝐴 ∖ (1...𝑁))) ∈ ℤ)
30		nn0z 12638	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
31	30	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℤ)
32		fzen 13581	. . . . . . . 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 1373	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...(♯‘(𝐴 ∖ (1...𝑁)))) ≈ ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)))
34	33	ensymd 9045	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)) ≈ (1...(♯‘(𝐴 ∖ (1...𝑁)))))
35		fzfi 14013	. . . . . . 7 ⊢ ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)) ∈ Fin
36		fzfi 14013	. . . . . . 7 ⊢ (1...(♯‘(𝐴 ∖ (1...𝑁)))) ∈ Fin
37		hashen 14386	. . . . . . 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 692	. . . . . 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 14385	. . . . . 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 2781	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((𝑁 + 1)...(♯‘𝐴))) = (♯‘(𝐴 ∖ (1...𝑁))))
43		fzfi 14013	. . . . 5 ⊢ ((𝑁 + 1)...(♯‘𝐴)) ∈ Fin
44		hashen 14386	. . . . 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 587	. . . 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 8995	. . 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 1192	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℕ0)
50	49	nn0zd 12639	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℤ)
51		simpl2 1193	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝐴 ∈ Fin)
52		hashcl 14395	. . . . . 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 12639	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (♯‘𝐴) ∈ ℤ)
55		nn0addge2 12573	. . . . . . 7 ⊢ ((𝑁 ∈ ℝ ∧ (♯‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0) → 𝑁 ≤ ((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))
56	2, 28, 55	syl2anc 584	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ≤ ((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))
57	56, 23	breqtrrd 5171	. . . . 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 12884	. . . 4 ⊢ ((♯‘𝐴) ∈ (ℤ≥‘𝑁) ↔ (𝑁 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑁 ≤ (♯‘𝐴)))
60	50, 54, 58, 59	syl3anbrc 1344	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (♯‘𝐴) ∈ (ℤ≥‘𝑁))
61		vex 3484	. . . . 5 ⊢ 𝑎 ∈ V
62		ovex 7464	. . . . . 6 ⊢ (1...𝑁) ∈ V
63		resiexg 7934	. . . . . 6 ⊢ ((1...𝑁) ∈ V → ( I ↾ (1...𝑁)) ∈ V)
64	62, 63	ax-mp 5	. . . . 5 ⊢ ( I ↾ (1...𝑁)) ∈ V
65	61, 64	unex 7764	. . . 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 6886	. . . . . 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 4209	. . . . . 6 ⊢ (((𝑁 + 1)...(♯‘𝐴)) ∩ (1...𝑁)) = ((1...𝑁) ∩ ((𝑁 + 1)...(♯‘𝐴)))
71	49	nn0red 12588	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℝ)
72	71	ltp1d 12198	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 < (𝑁 + 1))
73		fzdisj 13591	. . . . . . 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 2789	. . . . 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 6867	. . . . 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 4158	. . . . . 6 ⊢ (((𝑁 + 1)...(♯‘𝐴)) ∪ (1...𝑁)) = ((1...𝑁) ∪ ((𝑁 + 1)...(♯‘𝐴)))
80		fzsplit1nn0 42765	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (♯‘𝐴)) → (1...(♯‘𝐴)) = ((1...𝑁) ∪ ((𝑁 + 1)...(♯‘𝐴))))
81	49, 53, 58, 80	syl3anc 1373	. . . . . 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 2796	. . . . 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 1194	. . . . . . 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 2789	. . . . 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 6839	. . . . 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 584	. . . 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 6012	. . . 4 ⊢ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ((𝑎 ↾ (1...𝑁)) ∪ (( I ↾ (1...𝑁)) ↾ (1...𝑁)))
90		dmres 6030	. . . . . . . 8 ⊢ dom (𝑎 ↾ (1...𝑁)) = ((1...𝑁) ∩ dom 𝑎)
91		f1odm 6852	. . . . . . . . . . 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 4220	. . . . . . . . 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 2777	. . . . . . . 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 2789	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → dom (𝑎 ↾ (1...𝑁)) = ∅)
96		relres 6023	. . . . . . . 8 ⊢ Rel (𝑎 ↾ (1...𝑁))
97		reldm0 5938	. . . . . . . 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 6028	. . . . . . 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 4169	. . . . 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 4158	. . . . . 6 ⊢ (∅ ∪ ( I ↾ (1...𝑁))) = (( I ↾ (1...𝑁)) ∪ ∅)
104		un0 4394	. . . . . 6 ⊢ (( I ↾ (1...𝑁)) ∪ ∅) = ( I ↾ (1...𝑁))
105	103, 104	eqtri 2765	. . . . 5 ⊢ (∅ ∪ ( I ↾ (1...𝑁))) = ( I ↾ (1...𝑁))
106	102, 105	eqtrdi 2793	. . . 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 2789	. . 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 7439	. . . . . 6 ⊢ (𝑑 = (♯‘𝐴) → (1...𝑑) = (1...(♯‘𝐴)))
109	108	f1oeq2d 6844	. . . . 5 ⊢ (𝑑 = (♯‘𝐴) → (𝑒:(1...𝑑)–1-1-onto→𝐴 ↔ 𝑒:(1...(♯‘𝐴))–1-1-onto→𝐴))
110	109	anbi1d 631	. . . 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 6836	. . . . 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 5991	. . . . . 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 632	. . . 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 3635	. . 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 1376	. 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 1935	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 1540  ∃wex 1779   ∈ wcel 2108  ∃wrex 3070  Vcvv 3480   ∖ cdif 3948   ∪ cun 3949   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333   class class class wbr 5143   I cid 5577  dom cdm 5685   ↾ cres 5687  Rel wrel 5690  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431   ≈ cen 8982  Fincfn 8985  ℂcc 11153  ℝcr 11154  1c1 11156   + caddc 11158   < clt 11295   ≤ cle 11296  ℕ₀cn0 12526  ℤcz 12613  ℤ_≥cuz 12878  ...cfz 13547  ♯chash 14369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-hash 14370

This theorem is referenced by:  eldioph2  42773