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Theorem eldioph2lem1 36341
Description: Lemma for eldioph2 36343. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)
Assertion
Ref Expression
eldioph2lem1 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ∃𝑑 ∈ (ℤ𝑁)∃𝑒 ∈ V (𝑒:(1...𝑑)–1-1-onto𝐴 ∧ (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.
StepHypRef Expression
1 nn0re 11178 . . . . . . . . . 10 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
213ad2ant1 1075 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℝ)
32recnd 9947 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℂ)
4 ax-1cn 9873 . . . . . . . 8 1 ∈ ℂ
5 addcom 10101 . . . . . . . 8 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 + 1) = (1 + 𝑁))
63, 4, 5sylancl 693 . . . . . . 7 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (𝑁 + 1) = (1 + 𝑁))
7 diffi 8077 . . . . . . . . . 10 (𝐴 ∈ Fin → (𝐴 ∖ (1...𝑁)) ∈ Fin)
873ad2ant2 1076 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (𝐴 ∖ (1...𝑁)) ∈ Fin)
9 fzfid 12634 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...𝑁) ∈ Fin)
10 incom 3767 . . . . . . . . . . 11 ((𝐴 ∖ (1...𝑁)) ∩ (1...𝑁)) = ((1...𝑁) ∩ (𝐴 ∖ (1...𝑁)))
11 disjdif 3992 . . . . . . . . . . 11 ((1...𝑁) ∩ (𝐴 ∖ (1...𝑁))) = ∅
1210, 11eqtri 2632 . . . . . . . . . 10 ((𝐴 ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅
1312a1i 11 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝐴 ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅)
14 hashun 13032 . . . . . . . . 9 (((𝐴 ∖ (1...𝑁)) ∈ Fin ∧ (1...𝑁) ∈ Fin ∧ ((𝐴 ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅) → (#‘((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁))) = ((#‘(𝐴 ∖ (1...𝑁))) + (#‘(1...𝑁))))
158, 9, 13, 14syl3anc 1318 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁))) = ((#‘(𝐴 ∖ (1...𝑁))) + (#‘(1...𝑁))))
16 uncom 3719 . . . . . . . . . 10 ((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁)) = ((1...𝑁) ∪ (𝐴 ∖ (1...𝑁)))
17 simp3 1056 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...𝑁) ⊆ 𝐴)
18 undif 4001 . . . . . . . . . . 11 ((1...𝑁) ⊆ 𝐴 ↔ ((1...𝑁) ∪ (𝐴 ∖ (1...𝑁))) = 𝐴)
1917, 18sylib 207 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((1...𝑁) ∪ (𝐴 ∖ (1...𝑁))) = 𝐴)
2016, 19syl5eq 2656 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁)) = 𝐴)
2120fveq2d 6107 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁))) = (#‘𝐴))
22 hashfz1 12996 . . . . . . . . . 10 (𝑁 ∈ ℕ0 → (#‘(1...𝑁)) = 𝑁)
23223ad2ant1 1075 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘(1...𝑁)) = 𝑁)
2423oveq2d 6565 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((#‘(𝐴 ∖ (1...𝑁))) + (#‘(1...𝑁))) = ((#‘(𝐴 ∖ (1...𝑁))) + 𝑁))
2515, 21, 243eqtr3d 2652 . . . . . . 7 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘𝐴) = ((#‘(𝐴 ∖ (1...𝑁))) + 𝑁))
266, 25oveq12d 6567 . . . . . 6 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝑁 + 1)...(#‘𝐴)) = ((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁)))
2726fveq2d 6107 . . . . 5 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘((𝑁 + 1)...(#‘𝐴))) = (#‘((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁))))
28 1zzd 11285 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 1 ∈ ℤ)
29 hashcl 13009 . . . . . . . . . 10 ((𝐴 ∖ (1...𝑁)) ∈ Fin → (#‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0)
308, 29syl 17 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0)
3130nn0zd 11356 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘(𝐴 ∖ (1...𝑁))) ∈ ℤ)
32 nn0z 11277 . . . . . . . . 9 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
33323ad2ant1 1075 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℤ)
34 fzen 12229 . . . . . . . 8 ((1 ∈ ℤ ∧ (#‘(𝐴 ∖ (1...𝑁))) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (1...(#‘(𝐴 ∖ (1...𝑁)))) ≈ ((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁)))
3528, 31, 33, 34syl3anc 1318 . . . . . . 7 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...(#‘(𝐴 ∖ (1...𝑁)))) ≈ ((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁)))
3635ensymd 7893 . . . . . 6 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁)) ≈ (1...(#‘(𝐴 ∖ (1...𝑁)))))
37 fzfi 12633 . . . . . . 7 ((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁)) ∈ Fin
38 fzfi 12633 . . . . . . 7 (1...(#‘(𝐴 ∖ (1...𝑁)))) ∈ Fin
39 hashen 12997 . . . . . . 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))))))
4037, 38, 39mp2an 704 . . . . . 6 ((#‘((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁))) = (#‘(1...(#‘(𝐴 ∖ (1...𝑁))))) ↔ ((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁)) ≈ (1...(#‘(𝐴 ∖ (1...𝑁)))))
4136, 40sylibr 223 . . . . 5 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁))) = (#‘(1...(#‘(𝐴 ∖ (1...𝑁))))))
42 hashfz1 12996 . . . . . 6 ((#‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0 → (#‘(1...(#‘(𝐴 ∖ (1...𝑁))))) = (#‘(𝐴 ∖ (1...𝑁))))
4330, 42syl 17 . . . . 5 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘(1...(#‘(𝐴 ∖ (1...𝑁))))) = (#‘(𝐴 ∖ (1...𝑁))))
4427, 41, 433eqtrd 2648 . . . 4 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘((𝑁 + 1)...(#‘𝐴))) = (#‘(𝐴 ∖ (1...𝑁))))
45 fzfi 12633 . . . . 5 ((𝑁 + 1)...(#‘𝐴)) ∈ Fin
46 hashen 12997 . . . . 5 ((((𝑁 + 1)...(#‘𝐴)) ∈ Fin ∧ (𝐴 ∖ (1...𝑁)) ∈ Fin) → ((#‘((𝑁 + 1)...(#‘𝐴))) = (#‘(𝐴 ∖ (1...𝑁))) ↔ ((𝑁 + 1)...(#‘𝐴)) ≈ (𝐴 ∖ (1...𝑁))))
4745, 8, 46sylancr 694 . . . 4 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((#‘((𝑁 + 1)...(#‘𝐴))) = (#‘(𝐴 ∖ (1...𝑁))) ↔ ((𝑁 + 1)...(#‘𝐴)) ≈ (𝐴 ∖ (1...𝑁))))
4844, 47mpbid 221 . . 3 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝑁 + 1)...(#‘𝐴)) ≈ (𝐴 ∖ (1...𝑁)))
49 bren 7850 . . 3 (((𝑁 + 1)...(#‘𝐴)) ≈ (𝐴 ∖ (1...𝑁)) ↔ ∃𝑎 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁)))
5048, 49sylib 207 . 2 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ∃𝑎 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁)))
51 simpl1 1057 . . . . 5 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℕ0)
5251nn0zd 11356 . . . 4 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℤ)
53 simpl2 1058 . . . . . 6 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝐴 ∈ Fin)
54 hashcl 13009 . . . . . 6 (𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ0)
5553, 54syl 17 . . . . 5 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (#‘𝐴) ∈ ℕ0)
5655nn0zd 11356 . . . 4 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (#‘𝐴) ∈ ℤ)
57 nn0addge2 11217 . . . . . . 7 ((𝑁 ∈ ℝ ∧ (#‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0) → 𝑁 ≤ ((#‘(𝐴 ∖ (1...𝑁))) + 𝑁))
582, 30, 57syl2anc 691 . . . . . 6 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ≤ ((#‘(𝐴 ∖ (1...𝑁))) + 𝑁))
5958, 25breqtrrd 4611 . . . . 5 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ≤ (#‘𝐴))
6059adantr 480 . . . 4 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ≤ (#‘𝐴))
61 eluz2 11569 . . . 4 ((#‘𝐴) ∈ (ℤ𝑁) ↔ (𝑁 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ 𝑁 ≤ (#‘𝐴)))
6252, 56, 60, 61syl3anbrc 1239 . . 3 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (#‘𝐴) ∈ (ℤ𝑁))
63 vex 3176 . . . . 5 𝑎 ∈ V
64 ovex 6577 . . . . . 6 (1...𝑁) ∈ V
65 resiexg 6994 . . . . . 6 ((1...𝑁) ∈ V → ( I ↾ (1...𝑁)) ∈ V)
6664, 65ax-mp 5 . . . . 5 ( I ↾ (1...𝑁)) ∈ V
6763, 66unex 6854 . . . 4 (𝑎 ∪ ( I ↾ (1...𝑁))) ∈ V
6867a1i 11 . . 3 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (𝑎 ∪ ( I ↾ (1...𝑁))) ∈ V)
69 simpr 476 . . . . 5 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁)))
70 f1oi 6086 . . . . . 6 ( I ↾ (1...𝑁)):(1...𝑁)–1-1-onto→(1...𝑁)
7170a1i 11 . . . . 5 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ( I ↾ (1...𝑁)):(1...𝑁)–1-1-onto→(1...𝑁))
72 incom 3767 . . . . . 6 (((𝑁 + 1)...(#‘𝐴)) ∩ (1...𝑁)) = ((1...𝑁) ∩ ((𝑁 + 1)...(#‘𝐴)))
7351nn0red 11229 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℝ)
7473ltp1d 10833 . . . . . . 7 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 < (𝑁 + 1))
75 fzdisj 12239 . . . . . . 7 (𝑁 < (𝑁 + 1) → ((1...𝑁) ∩ ((𝑁 + 1)...(#‘𝐴))) = ∅)
7674, 75syl 17 . . . . . 6 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ((1...𝑁) ∩ ((𝑁 + 1)...(#‘𝐴))) = ∅)
7772, 76syl5eq 2656 . . . . 5 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (((𝑁 + 1)...(#‘𝐴)) ∩ (1...𝑁)) = ∅)
7812a1i 11 . . . . 5 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ((𝐴 ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅)
79 f1oun 6069 . . . . 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)))
8069, 71, 77, 78, 79syl22anc 1319 . . . 4 (((𝑁 ∈ ℕ0𝐴 ∈ 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)))
81 fzsplit1nn0 36335 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0𝑁 ≤ (#‘𝐴)) → (1...(#‘𝐴)) = ((1...𝑁) ∪ ((𝑁 + 1)...(#‘𝐴))))
8251, 55, 60, 81syl3anc 1318 . . . . . 6 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (1...(#‘𝐴)) = ((1...𝑁) ∪ ((𝑁 + 1)...(#‘𝐴))))
83 uncom 3719 . . . . . 6 (((𝑁 + 1)...(#‘𝐴)) ∪ (1...𝑁)) = ((1...𝑁) ∪ ((𝑁 + 1)...(#‘𝐴)))
8482, 83syl6reqr 2663 . . . . 5 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (((𝑁 + 1)...(#‘𝐴)) ∪ (1...𝑁)) = (1...(#‘𝐴)))
85 simpl3 1059 . . . . . . 7 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (1...𝑁) ⊆ 𝐴)
8685, 18sylib 207 . . . . . 6 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ((1...𝑁) ∪ (𝐴 ∖ (1...𝑁))) = 𝐴)
8716, 86syl5eq 2656 . . . . 5 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁)) = 𝐴)
88 f1oeq23 6043 . . . . 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-onto𝐴))
8984, 87, 88syl2anc 691 . . . 4 (((𝑁 ∈ ℕ0𝐴 ∈ 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-onto𝐴))
9080, 89mpbid 221 . . 3 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (𝑎 ∪ ( I ↾ (1...𝑁))):(1...(#‘𝐴))–1-1-onto𝐴)
91 resundir 5331 . . . 4 ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ((𝑎 ↾ (1...𝑁)) ∪ (( I ↾ (1...𝑁)) ↾ (1...𝑁)))
92 dmres 5339 . . . . . . . 8 dom (𝑎 ↾ (1...𝑁)) = ((1...𝑁) ∩ dom 𝑎)
93 f1odm 6054 . . . . . . . . . . 11 (𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁)) → dom 𝑎 = ((𝑁 + 1)...(#‘𝐴)))
9493adantl 481 . . . . . . . . . 10 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → dom 𝑎 = ((𝑁 + 1)...(#‘𝐴)))
9594ineq2d 3776 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ((1...𝑁) ∩ dom 𝑎) = ((1...𝑁) ∩ ((𝑁 + 1)...(#‘𝐴))))
9695, 76eqtrd 2644 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ((1...𝑁) ∩ dom 𝑎) = ∅)
9792, 96syl5eq 2656 . . . . . . 7 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → dom (𝑎 ↾ (1...𝑁)) = ∅)
98 relres 5346 . . . . . . . 8 Rel (𝑎 ↾ (1...𝑁))
99 reldm0 5264 . . . . . . . 8 (Rel (𝑎 ↾ (1...𝑁)) → ((𝑎 ↾ (1...𝑁)) = ∅ ↔ dom (𝑎 ↾ (1...𝑁)) = ∅))
10098, 99ax-mp 5 . . . . . . 7 ((𝑎 ↾ (1...𝑁)) = ∅ ↔ dom (𝑎 ↾ (1...𝑁)) = ∅)
10197, 100sylibr 223 . . . . . 6 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (𝑎 ↾ (1...𝑁)) = ∅)
102 residm 5350 . . . . . . 7 (( I ↾ (1...𝑁)) ↾ (1...𝑁)) = ( I ↾ (1...𝑁))
103102a1i 11 . . . . . 6 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (( I ↾ (1...𝑁)) ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))
104101, 103uneq12d 3730 . . . . 5 (((𝑁 ∈ ℕ0𝐴 ∈ 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))))
105 uncom 3719 . . . . . 6 (∅ ∪ ( I ↾ (1...𝑁))) = (( I ↾ (1...𝑁)) ∪ ∅)
106 un0 3919 . . . . . 6 (( I ↾ (1...𝑁)) ∪ ∅) = ( I ↾ (1...𝑁))
107105, 106eqtri 2632 . . . . 5 (∅ ∪ ( I ↾ (1...𝑁))) = ( I ↾ (1...𝑁))
108104, 107syl6eq 2660 . . . 4 (((𝑁 ∈ ℕ0𝐴 ∈ 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)))
10991, 108syl5eq 2656 . . 3 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))
110 oveq2 6557 . . . . . 6 (𝑑 = (#‘𝐴) → (1...𝑑) = (1...(#‘𝐴)))
111 f1oeq2 6041 . . . . . 6 ((1...𝑑) = (1...(#‘𝐴)) → (𝑒:(1...𝑑)–1-1-onto𝐴𝑒:(1...(#‘𝐴))–1-1-onto𝐴))
112110, 111syl 17 . . . . 5 (𝑑 = (#‘𝐴) → (𝑒:(1...𝑑)–1-1-onto𝐴𝑒:(1...(#‘𝐴))–1-1-onto𝐴))
113112anbi1d 737 . . . 4 (𝑑 = (#‘𝐴) → ((𝑒:(1...𝑑)–1-1-onto𝐴 ∧ (𝑒 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) ↔ (𝑒:(1...(#‘𝐴))–1-1-onto𝐴 ∧ (𝑒 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))))
114 f1oeq1 6040 . . . . 5 (𝑒 = (𝑎 ∪ ( I ↾ (1...𝑁))) → (𝑒:(1...(#‘𝐴))–1-1-onto𝐴 ↔ (𝑎 ∪ ( I ↾ (1...𝑁))):(1...(#‘𝐴))–1-1-onto𝐴))
115 reseq1 5311 . . . . . 6 (𝑒 = (𝑎 ∪ ( I ↾ (1...𝑁))) → (𝑒 ↾ (1...𝑁)) = ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)))
116115eqeq1d 2612 . . . . 5 (𝑒 = (𝑎 ∪ ( I ↾ (1...𝑁))) → ((𝑒 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)) ↔ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
117114, 116anbi12d 743 . . . 4 (𝑒 = (𝑎 ∪ ( I ↾ (1...𝑁))) → ((𝑒:(1...(#‘𝐴))–1-1-onto𝐴 ∧ (𝑒 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) ↔ ((𝑎 ∪ ( I ↾ (1...𝑁))):(1...(#‘𝐴))–1-1-onto𝐴 ∧ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))))
118113, 117rspc2ev 3295 . . 3 (((#‘𝐴) ∈ (ℤ𝑁) ∧ (𝑎 ∪ ( I ↾ (1...𝑁))) ∈ V ∧ ((𝑎 ∪ ( I ↾ (1...𝑁))):(1...(#‘𝐴))–1-1-onto𝐴 ∧ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → ∃𝑑 ∈ (ℤ𝑁)∃𝑒 ∈ V (𝑒:(1...𝑑)–1-1-onto𝐴 ∧ (𝑒 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
11962, 68, 90, 109, 118syl112anc 1322 . 2 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ∃𝑑 ∈ (ℤ𝑁)∃𝑒 ∈ V (𝑒:(1...𝑑)–1-1-onto𝐴 ∧ (𝑒 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
12050, 119exlimddv 1850 1 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ∃𝑑 ∈ (ℤ𝑁)∃𝑒 ∈ V (𝑒:(1...𝑑)–1-1-onto𝐴 ∧ (𝑒 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wex 1695  wcel 1977  wrex 2897  Vcvv 3173  cdif 3537  cun 3538  cin 3539  wss 3540  c0 3874   class class class wbr 4583   I cid 4948  dom cdm 5038  cres 5040  Rel wrel 5043  1-1-ontowf1o 5803  cfv 5804  (class class class)co 6549  cen 7838  Fincfn 7841  cc 9813  cr 9814  1c1 9816   + caddc 9818   < clt 9953  cle 9954  0cn0 11169  cz 11254  cuz 11563  ...cfz 12197  #chash 12979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980
This theorem is referenced by:  eldioph2  36343
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