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Theorem eldioph2lem1 36838
 Description: Lemma for eldioph2 36840. 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 11253 . . . . . . . . . 10 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
213ad2ant1 1080 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℝ)
32recnd 10020 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℂ)
4 ax-1cn 9946 . . . . . . . 8 1 ∈ ℂ
5 addcom 10174 . . . . . . . 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 8144 . . . . . . . . . 10 (𝐴 ∈ Fin → (𝐴 ∖ (1...𝑁)) ∈ Fin)
873ad2ant2 1081 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (𝐴 ∖ (1...𝑁)) ∈ Fin)
9 fzfid 12720 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...𝑁) ∈ Fin)
10 incom 3788 . . . . . . . . . . 11 ((𝐴 ∖ (1...𝑁)) ∩ (1...𝑁)) = ((1...𝑁) ∩ (𝐴 ∖ (1...𝑁)))
11 disjdif 4017 . . . . . . . . . . 11 ((1...𝑁) ∩ (𝐴 ∖ (1...𝑁))) = ∅
1210, 11eqtri 2643 . . . . . . . . . 10 ((𝐴 ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅
1312a1i 11 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝐴 ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅)
14 hashun 13119 . . . . . . . . 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 1323 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁))) = ((#‘(𝐴 ∖ (1...𝑁))) + (#‘(1...𝑁))))
16 uncom 3740 . . . . . . . . . 10 ((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁)) = ((1...𝑁) ∪ (𝐴 ∖ (1...𝑁)))
17 simp3 1061 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...𝑁) ⊆ 𝐴)
18 undif 4026 . . . . . . . . . . 11 ((1...𝑁) ⊆ 𝐴 ↔ ((1...𝑁) ∪ (𝐴 ∖ (1...𝑁))) = 𝐴)
1917, 18sylib 208 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((1...𝑁) ∪ (𝐴 ∖ (1...𝑁))) = 𝐴)
2016, 19syl5eq 2667 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁)) = 𝐴)
2120fveq2d 6157 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁))) = (#‘𝐴))
22 hashfz1 13082 . . . . . . . . . 10 (𝑁 ∈ ℕ0 → (#‘(1...𝑁)) = 𝑁)
23223ad2ant1 1080 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘(1...𝑁)) = 𝑁)
2423oveq2d 6626 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((#‘(𝐴 ∖ (1...𝑁))) + (#‘(1...𝑁))) = ((#‘(𝐴 ∖ (1...𝑁))) + 𝑁))
2515, 21, 243eqtr3d 2663 . . . . . . 7 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘𝐴) = ((#‘(𝐴 ∖ (1...𝑁))) + 𝑁))
266, 25oveq12d 6628 . . . . . 6 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝑁 + 1)...(#‘𝐴)) = ((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁)))
2726fveq2d 6157 . . . . 5 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘((𝑁 + 1)...(#‘𝐴))) = (#‘((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁))))
28 1zzd 11360 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 1 ∈ ℤ)
29 hashcl 13095 . . . . . . . . . 10 ((𝐴 ∖ (1...𝑁)) ∈ Fin → (#‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0)
308, 29syl 17 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0)
3130nn0zd 11432 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘(𝐴 ∖ (1...𝑁))) ∈ ℤ)
32 nn0z 11352 . . . . . . . . 9 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
33323ad2ant1 1080 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℤ)
34 fzen 12308 . . . . . . . 8 ((1 ∈ ℤ ∧ (#‘(𝐴 ∖ (1...𝑁))) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (1...(#‘(𝐴 ∖ (1...𝑁)))) ≈ ((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁)))
3528, 31, 33, 34syl3anc 1323 . . . . . . 7 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...(#‘(𝐴 ∖ (1...𝑁)))) ≈ ((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁)))
3635ensymd 7959 . . . . . 6 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁)) ≈ (1...(#‘(𝐴 ∖ (1...𝑁)))))
37 fzfi 12719 . . . . . . 7 ((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁)) ∈ Fin
38 fzfi 12719 . . . . . . 7 (1...(#‘(𝐴 ∖ (1...𝑁)))) ∈ Fin
39 hashen 13083 . . . . . . 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 707 . . . . . 6 ((#‘((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁))) = (#‘(1...(#‘(𝐴 ∖ (1...𝑁))))) ↔ ((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁)) ≈ (1...(#‘(𝐴 ∖ (1...𝑁)))))
4136, 40sylibr 224 . . . . 5 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁))) = (#‘(1...(#‘(𝐴 ∖ (1...𝑁))))))
42 hashfz1 13082 . . . . . 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 2659 . . . 4 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘((𝑁 + 1)...(#‘𝐴))) = (#‘(𝐴 ∖ (1...𝑁))))
45 fzfi 12719 . . . . 5 ((𝑁 + 1)...(#‘𝐴)) ∈ Fin
46 hashen 13083 . . . . 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 222 . . 3 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝑁 + 1)...(#‘𝐴)) ≈ (𝐴 ∖ (1...𝑁)))
49 bren 7916 . . 3 (((𝑁 + 1)...(#‘𝐴)) ≈ (𝐴 ∖ (1...𝑁)) ↔ ∃𝑎 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁)))
5048, 49sylib 208 . 2 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ∃𝑎 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁)))
51 simpl1 1062 . . . . 5 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℕ0)
5251nn0zd 11432 . . . 4 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℤ)
53 simpl2 1063 . . . . . 6 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝐴 ∈ Fin)
54 hashcl 13095 . . . . . 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 11432 . . . 4 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (#‘𝐴) ∈ ℤ)
57 nn0addge2 11292 . . . . . . 7 ((𝑁 ∈ ℝ ∧ (#‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0) → 𝑁 ≤ ((#‘(𝐴 ∖ (1...𝑁))) + 𝑁))
582, 30, 57syl2anc 692 . . . . . 6 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ≤ ((#‘(𝐴 ∖ (1...𝑁))) + 𝑁))
5958, 25breqtrrd 4646 . . . . 5 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ≤ (#‘𝐴))
6059adantr 481 . . . 4 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ≤ (#‘𝐴))
61 eluz2 11645 . . . 4 ((#‘𝐴) ∈ (ℤ𝑁) ↔ (𝑁 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ 𝑁 ≤ (#‘𝐴)))
6252, 56, 60, 61syl3anbrc 1244 . . 3 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (#‘𝐴) ∈ (ℤ𝑁))
63 vex 3192 . . . . 5 𝑎 ∈ V
64 ovex 6638 . . . . . 6 (1...𝑁) ∈ V
65 resiexg 7056 . . . . . 6 ((1...𝑁) ∈ V → ( I ↾ (1...𝑁)) ∈ V)
6664, 65ax-mp 5 . . . . 5 ( I ↾ (1...𝑁)) ∈ V
6763, 66unex 6916 . . . 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 477 . . . . 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 6136 . . . . . 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 3788 . . . . . 6 (((𝑁 + 1)...(#‘𝐴)) ∩ (1...𝑁)) = ((1...𝑁) ∩ ((𝑁 + 1)...(#‘𝐴)))
7351nn0red 11304 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℝ)
7473ltp1d 10906 . . . . . . 7 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 < (𝑁 + 1))
75 fzdisj 12318 . . . . . . 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 2667 . . . . 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 6118 . . . . 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 1324 . . . 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 36832 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0𝑁 ≤ (#‘𝐴)) → (1...(#‘𝐴)) = ((1...𝑁) ∪ ((𝑁 + 1)...(#‘𝐴))))
8251, 55, 60, 81syl3anc 1323 . . . . . 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 3740 . . . . . 6 (((𝑁 + 1)...(#‘𝐴)) ∪ (1...𝑁)) = ((1...𝑁) ∪ ((𝑁 + 1)...(#‘𝐴)))
8482, 83syl6reqr 2674 . . . . 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 1064 . . . . . . 7 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (1...𝑁) ⊆ 𝐴)
8685, 18sylib 208 . . . . . 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 2667 . . . . 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 6092 . . . . 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 692 . . . 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 222 . . 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 5375 . . . 4 ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ((𝑎 ↾ (1...𝑁)) ∪ (( I ↾ (1...𝑁)) ↾ (1...𝑁)))
92 dmres 5383 . . . . . . . 8 dom (𝑎 ↾ (1...𝑁)) = ((1...𝑁) ∩ dom 𝑎)
93 f1odm 6103 . . . . . . . . . . 11 (𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁)) → dom 𝑎 = ((𝑁 + 1)...(#‘𝐴)))
9493adantl 482 . . . . . . . . . 10 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → dom 𝑎 = ((𝑁 + 1)...(#‘𝐴)))
9594ineq2d 3797 . . . . . . . . 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 2655 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ((1...𝑁) ∩ dom 𝑎) = ∅)
9792, 96syl5eq 2667 . . . . . . 7 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → dom (𝑎 ↾ (1...𝑁)) = ∅)
98 relres 5390 . . . . . . . 8 Rel (𝑎 ↾ (1...𝑁))
99 reldm0 5308 . . . . . . . 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 224 . . . . . 6 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (𝑎 ↾ (1...𝑁)) = ∅)
102 residm 5394 . . . . . . 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 3751 . . . . 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 3740 . . . . . 6 (∅ ∪ ( I ↾ (1...𝑁))) = (( I ↾ (1...𝑁)) ∪ ∅)
106 un0 3944 . . . . . 6 (( I ↾ (1...𝑁)) ∪ ∅) = ( I ↾ (1...𝑁))
107105, 106eqtri 2643 . . . . 5 (∅ ∪ ( I ↾ (1...𝑁))) = ( I ↾ (1...𝑁))
108104, 107syl6eq 2671 . . . 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 2667 . . 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 6618 . . . . . 6 (𝑑 = (#‘𝐴) → (1...𝑑) = (1...(#‘𝐴)))
111 f1oeq2 6090 . . . . . 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 740 . . . 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 6089 . . . . 5 (𝑒 = (𝑎 ∪ ( I ↾ (1...𝑁))) → (𝑒:(1...(#‘𝐴))–1-1-onto𝐴 ↔ (𝑎 ∪ ( I ↾ (1...𝑁))):(1...(#‘𝐴))–1-1-onto𝐴))
115 reseq1 5355 . . . . . 6 (𝑒 = (𝑎 ∪ ( I ↾ (1...𝑁))) → (𝑒 ↾ (1...𝑁)) = ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)))
116115eqeq1d 2623 . . . . 5 (𝑒 = (𝑎 ∪ ( I ↾ (1...𝑁))) → ((𝑒 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)) ↔ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
117114, 116anbi12d 746 . . . 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 3312 . . 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 1327 . 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 1860 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 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480  ∃wex 1701   ∈ wcel 1987  ∃wrex 2908  Vcvv 3189   ∖ cdif 3556   ∪ cun 3557   ∩ cin 3558   ⊆ wss 3559  ∅c0 3896   class class class wbr 4618   I cid 4989  dom cdm 5079   ↾ cres 5081  Rel wrel 5084  –1-1-onto→wf1o 5851  ‘cfv 5852  (class class class)co 6610   ≈ cen 7904  Fincfn 7907  ℂcc 9886  ℝcr 9887  1c1 9889   + caddc 9891   < clt 10026   ≤ cle 10027  ℕ0cn0 11244  ℤcz 11329  ℤ≥cuz 11639  ...cfz 12276  #chash 13065 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-card 8717  df-cda 8942  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-n0 11245  df-z 11330  df-uz 11640  df-fz 12277  df-hash 13066 This theorem is referenced by:  eldioph2  36840
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