Theorem eldioph2lem1 43418

Description: Lemma for eldioph2 43420. 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 12513	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
2	1	3ad2ant1 1149	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℝ)
3	2	recnd 11237	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℂ)
4		ax-1cn 11158	. . . . . . . 8 ⊢ 1 ∈ ℂ
5		addcom 11396	. . . . . . . 8 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 + 1) = (1 + 𝑁))
6	3, 4, 5	sylancl 597	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (𝑁 + 1) = (1 + 𝑁))
7		diffi 9159	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ (1...𝑁)) ∈ Fin)
8	7	3ad2ant2 1150	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (𝐴 ∖ (1...𝑁)) ∈ Fin)
9		fzfid 14009	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...𝑁) ∈ Fin)
10		disjdifr 4437	. . . . . . . . . 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 14418	. . . . . . . . 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 1396	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁))) = ((♯‘(𝐴 ∖ (1...𝑁))) + (♯‘(1...𝑁))))
14		uncom 4118	. . . . . . . . . 10 ⊢ ((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁)) = ((1...𝑁) ∪ (𝐴 ∖ (1...𝑁)))
15		simp3 1154	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...𝑁) ⊆ 𝐴)
16		undif 4446	. . . . . . . . . . 11 ⊢ ((1...𝑁) ⊆ 𝐴 ↔ ((1...𝑁) ∪ (𝐴 ∖ (1...𝑁))) = 𝐴)
17	15, 16	sylib 221	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((1...𝑁) ∪ (𝐴 ∖ (1...𝑁))) = 𝐴)
18	14, 17	eqtrid 2816	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁)) = 𝐴)
19	18	fveq2d 6886	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁))) = (♯‘𝐴))
20		hashfz1 14382	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
21	20	3ad2ant1 1149	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘(1...𝑁)) = 𝑁)
22	21	oveq2d 7427	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((♯‘(𝐴 ∖ (1...𝑁))) + (♯‘(1...𝑁))) = ((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))
23	13, 19, 22	3eqtr3d 2812	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘𝐴) = ((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))
24	6, 23	oveq12d 7429	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝑁 + 1)...(♯‘𝐴)) = ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)))
25	24	fveq2d 6886	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((𝑁 + 1)...(♯‘𝐴))) = (♯‘((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))))
26		1zzd 12625	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 1 ∈ ℤ)
27		hashcl 14392	. . . . . . . . . 10 ⊢ ((𝐴 ∖ (1...𝑁)) ∈ Fin → (♯‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0)
28	8, 27	syl 18	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0)
29	28	nn0zd 12616	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘(𝐴 ∖ (1...𝑁))) ∈ ℤ)
30		nn0z 12615	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
31	30	3ad2ant1 1149	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℤ)
32		fzen 13569	. . . . . . . 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 1396	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...(♯‘(𝐴 ∖ (1...𝑁)))) ≈ ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)))
34	33	ensymd 9002	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)) ≈ (1...(♯‘(𝐴 ∖ (1...𝑁)))))
35		fzfi 14008	. . . . . . 7 ⊢ ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)) ∈ Fin
36		fzfi 14008	. . . . . . 7 ⊢ (1...(♯‘(𝐴 ∖ (1...𝑁)))) ∈ Fin
37		hashen 14383	. . . . . . 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 704	. . . . . 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 237	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))) = (♯‘(1...(♯‘(𝐴 ∖ (1...𝑁))))))
40		hashfz1 14382	. . . . . 6 ⊢ ((♯‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0 → (♯‘(1...(♯‘(𝐴 ∖ (1...𝑁))))) = (♯‘(𝐴 ∖ (1...𝑁))))
41	28, 40	syl 18	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘(1...(♯‘(𝐴 ∖ (1...𝑁))))) = (♯‘(𝐴 ∖ (1...𝑁))))
42	25, 39, 41	3eqtrd 2808	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((𝑁 + 1)...(♯‘𝐴))) = (♯‘(𝐴 ∖ (1...𝑁))))
43		fzfi 14008	. . . . 5 ⊢ ((𝑁 + 1)...(♯‘𝐴)) ∈ Fin
44		hashen 14383	. . . . 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 598	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((♯‘((𝑁 + 1)...(♯‘𝐴))) = (♯‘(𝐴 ∖ (1...𝑁))) ↔ ((𝑁 + 1)...(♯‘𝐴)) ≈ (𝐴 ∖ (1...𝑁))))
46	42, 45	mpbid 235	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝑁 + 1)...(♯‘𝐴)) ≈ (𝐴 ∖ (1...𝑁)))
47		bren 8953	. . 3 ⊢ (((𝑁 + 1)...(♯‘𝐴)) ≈ (𝐴 ∖ (1...𝑁)) ↔ ∃𝑎 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁)))
48	46, 47	sylib 221	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ∃𝑎 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁)))
49		simpl1 1208	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℕ0)
50	49	nn0zd 12616	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℤ)
51		simpl2 1209	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝐴 ∈ Fin)
52		hashcl 14392	. . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
53	51, 52	syl 18	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (♯‘𝐴) ∈ ℕ0)
54	53	nn0zd 12616	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (♯‘𝐴) ∈ ℤ)
55		nn0addge2 12551	. . . . . . 7 ⊢ ((𝑁 ∈ ℝ ∧ (♯‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0) → 𝑁 ≤ ((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))
56	2, 28, 55	syl2anc 595	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ≤ ((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))
57	56, 23	breqtrrd 5141	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ≤ (♯‘𝐴))
58	57	adantr 485	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ≤ (♯‘𝐴))
59		eluz2 12868	. . . 4 ⊢ ((♯‘𝐴) ∈ (ℤ≥‘𝑁) ↔ (𝑁 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑁 ≤ (♯‘𝐴)))
60	50, 54, 58, 59	syl3anbrc 1360	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (♯‘𝐴) ∈ (ℤ≥‘𝑁))
61		vex 3465	. . . . 5 ⊢ 𝑎 ∈ V
62		ovex 7444	. . . . . 6 ⊢ (1...𝑁) ∈ V
63		resiexg 7909	. . . . . 6 ⊢ ((1...𝑁) ∈ V → ( I ↾ (1...𝑁)) ∈ V)
64	62, 63	ax-mp 5	. . . . 5 ⊢ ( I ↾ (1...𝑁)) ∈ V
65	61, 64	unex 7743	. . . 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 489	. . . . 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 6860	. . . . . 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 4168	. . . . . 6 ⊢ (((𝑁 + 1)...(♯‘𝐴)) ∩ (1...𝑁)) = ((1...𝑁) ∩ ((𝑁 + 1)...(♯‘𝐴)))
71	49	nn0red 12566	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℝ)
72	71	ltp1d 12145	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 < (𝑁 + 1))
73		fzdisj 13579	. . . . . . 7 ⊢ (𝑁 < (𝑁 + 1) → ((1...𝑁) ∩ ((𝑁 + 1)...(♯‘𝐴))) = ∅)
74	72, 73	syl 18	. . . . . 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 2816	. . . . 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 6841	. . . . 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 851	. . . 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 4118	. . . . . 6 ⊢ (((𝑁 + 1)...(♯‘𝐴)) ∪ (1...𝑁)) = ((1...𝑁) ∪ ((𝑁 + 1)...(♯‘𝐴)))
80		fzsplit1nn0 43412	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0 ∧ 𝑁 ≤ (♯‘𝐴)) → (1...(♯‘𝐴)) = ((1...𝑁) ∪ ((𝑁 + 1)...(♯‘𝐴))))
81	49, 53, 58, 80	syl3anc 1396	. . . . . 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 2823	. . . . 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 1210	. . . . . . 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 221	. . . . . 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 2816	. . . . 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 6812	. . . . 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 595	. . . 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 235	. . 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 5994	. . . 4 ⊢ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ((𝑎 ↾ (1...𝑁)) ∪ (( I ↾ (1...𝑁)) ↾ (1...𝑁)))
90		dmres 6012	. . . . . . . 8 ⊢ dom (𝑎 ↾ (1...𝑁)) = ((1...𝑁) ∩ dom 𝑎)
91		f1odm 6825	. . . . . . . . . . 11 ⊢ (𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁)) → dom 𝑎 = ((𝑁 + 1)...(♯‘𝐴)))
92	91	adantl 486	. . . . . . . . . 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 4179	. . . . . . . . 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 2804	. . . . . . . 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 2816	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → dom (𝑎 ↾ (1...𝑁)) = ∅)
96		relres 6005	. . . . . . . 8 ⊢ Rel (𝑎 ↾ (1...𝑁))
97		reldm0 5919	. . . . . . . 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 237	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (𝑎 ↾ (1...𝑁)) = ∅)
100		residm 6010	. . . . . . 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 4129	. . . . 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 4118	. . . . . 6 ⊢ (∅ ∪ ( I ↾ (1...𝑁))) = (( I ↾ (1...𝑁)) ∪ ∅)
104		un0 4356	. . . . . 6 ⊢ (( I ↾ (1...𝑁)) ∪ ∅) = ( I ↾ (1...𝑁))
105	103, 104	eqtri 2792	. . . . 5 ⊢ (∅ ∪ ( I ↾ (1...𝑁))) = ( I ↾ (1...𝑁))
106	102, 105	eqtrdi 2820	. . . 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 2816	. . 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 7419	. . . . . 6 ⊢ (𝑑 = (♯‘𝐴) → (1...𝑑) = (1...(♯‘𝐴)))
109	108	f1oeq2d 6817	. . . . 5 ⊢ (𝑑 = (♯‘𝐴) → (𝑒:(1...𝑑)–1-1-onto→𝐴 ↔ 𝑒:(1...(♯‘𝐴))–1-1-onto→𝐴))
110	109	anbi1d 642	. . . 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 6809	. . . . 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 5973	. . . . . 6 ⊢ (𝑒 = (𝑎 ∪ ( I ↾ (1...𝑁))) → (𝑒 ↾ (1...𝑁)) = ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)))
113	112	eqeq1d 2771	. . . . 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 643	. . . 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 3601	. . 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 1399	. 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 1962	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 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∃wrex 3095  Vcvv 3461   ∖ cdif 3908   ∪ cun 3909   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292   class class class wbr 5111   I cid 5556  dom cdm 5662   ↾ cres 5664  Rel wrel 5667  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7411   ≈ cen 8940  Fincfn 8943  ℂcc 11098  ℝcr 11099  1c1 11101   + caddc 11103   < clt 11243   ≤ cle 11244  ℕ₀cn0 12504  ℤcz 12591  ℤ_≥cuz 12862  ...cfz 13535  ♯chash 14366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-oadd 8457  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-dju 9887  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-n0 12505  df-z 12592  df-uz 12863  df-fz 13536  df-hash 14367

This theorem is referenced by:  eldioph2  43420