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Theorem eldioph2lem1 37640
Description: Lemma for eldioph2 37642. 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 11339 . . . . . . . . . 10 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
213ad2ant1 1102 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℝ)
32recnd 10106 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℂ)
4 ax-1cn 10032 . . . . . . . 8 1 ∈ ℂ
5 addcom 10260 . . . . . . . 8 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 + 1) = (1 + 𝑁))
63, 4, 5sylancl 695 . . . . . . 7 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (𝑁 + 1) = (1 + 𝑁))
7 diffi 8233 . . . . . . . . . 10 (𝐴 ∈ Fin → (𝐴 ∖ (1...𝑁)) ∈ Fin)
873ad2ant2 1103 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (𝐴 ∖ (1...𝑁)) ∈ Fin)
9 fzfid 12812 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...𝑁) ∈ Fin)
10 incom 3838 . . . . . . . . . . 11 ((𝐴 ∖ (1...𝑁)) ∩ (1...𝑁)) = ((1...𝑁) ∩ (𝐴 ∖ (1...𝑁)))
11 disjdif 4073 . . . . . . . . . . 11 ((1...𝑁) ∩ (𝐴 ∖ (1...𝑁))) = ∅
1210, 11eqtri 2673 . . . . . . . . . 10 ((𝐴 ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅
1312a1i 11 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝐴 ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅)
14 hashun 13209 . . . . . . . . 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 1366 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁))) = ((#‘(𝐴 ∖ (1...𝑁))) + (#‘(1...𝑁))))
16 uncom 3790 . . . . . . . . . 10 ((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁)) = ((1...𝑁) ∪ (𝐴 ∖ (1...𝑁)))
17 simp3 1083 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...𝑁) ⊆ 𝐴)
18 undif 4082 . . . . . . . . . . 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 2697 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁)) = 𝐴)
2120fveq2d 6233 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁))) = (#‘𝐴))
22 hashfz1 13174 . . . . . . . . . 10 (𝑁 ∈ ℕ0 → (#‘(1...𝑁)) = 𝑁)
23223ad2ant1 1102 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘(1...𝑁)) = 𝑁)
2423oveq2d 6706 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((#‘(𝐴 ∖ (1...𝑁))) + (#‘(1...𝑁))) = ((#‘(𝐴 ∖ (1...𝑁))) + 𝑁))
2515, 21, 243eqtr3d 2693 . . . . . . 7 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘𝐴) = ((#‘(𝐴 ∖ (1...𝑁))) + 𝑁))
266, 25oveq12d 6708 . . . . . 6 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝑁 + 1)...(#‘𝐴)) = ((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁)))
2726fveq2d 6233 . . . . 5 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘((𝑁 + 1)...(#‘𝐴))) = (#‘((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁))))
28 1zzd 11446 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 1 ∈ ℤ)
29 hashcl 13185 . . . . . . . . . 10 ((𝐴 ∖ (1...𝑁)) ∈ Fin → (#‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0)
308, 29syl 17 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0)
3130nn0zd 11518 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘(𝐴 ∖ (1...𝑁))) ∈ ℤ)
32 nn0z 11438 . . . . . . . . 9 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
33323ad2ant1 1102 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℤ)
34 fzen 12396 . . . . . . . 8 ((1 ∈ ℤ ∧ (#‘(𝐴 ∖ (1...𝑁))) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (1...(#‘(𝐴 ∖ (1...𝑁)))) ≈ ((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁)))
3528, 31, 33, 34syl3anc 1366 . . . . . . 7 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...(#‘(𝐴 ∖ (1...𝑁)))) ≈ ((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁)))
3635ensymd 8048 . . . . . 6 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁)) ≈ (1...(#‘(𝐴 ∖ (1...𝑁)))))
37 fzfi 12811 . . . . . . 7 ((1 + 𝑁)...((#‘(𝐴 ∖ (1...𝑁))) + 𝑁)) ∈ Fin
38 fzfi 12811 . . . . . . 7 (1...(#‘(𝐴 ∖ (1...𝑁)))) ∈ Fin
39 hashen 13175 . . . . . . 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 708 . . . . . 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 13174 . . . . . 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 2689 . . . 4 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (#‘((𝑁 + 1)...(#‘𝐴))) = (#‘(𝐴 ∖ (1...𝑁))))
45 fzfi 12811 . . . . 5 ((𝑁 + 1)...(#‘𝐴)) ∈ Fin
46 hashen 13175 . . . . 5 ((((𝑁 + 1)...(#‘𝐴)) ∈ Fin ∧ (𝐴 ∖ (1...𝑁)) ∈ Fin) → ((#‘((𝑁 + 1)...(#‘𝐴))) = (#‘(𝐴 ∖ (1...𝑁))) ↔ ((𝑁 + 1)...(#‘𝐴)) ≈ (𝐴 ∖ (1...𝑁))))
4745, 8, 46sylancr 696 . . . 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 8006 . . 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 1084 . . . . 5 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℕ0)
5251nn0zd 11518 . . . 4 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℤ)
53 simpl2 1085 . . . . . 6 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝐴 ∈ Fin)
54 hashcl 13185 . . . . . 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 11518 . . . 4 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (#‘𝐴) ∈ ℤ)
57 nn0addge2 11378 . . . . . . 7 ((𝑁 ∈ ℝ ∧ (#‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0) → 𝑁 ≤ ((#‘(𝐴 ∖ (1...𝑁))) + 𝑁))
582, 30, 57syl2anc 694 . . . . . 6 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ≤ ((#‘(𝐴 ∖ (1...𝑁))) + 𝑁))
5958, 25breqtrrd 4713 . . . . 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 11731 . . . 4 ((#‘𝐴) ∈ (ℤ𝑁) ↔ (𝑁 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ 𝑁 ≤ (#‘𝐴)))
6252, 56, 60, 61syl3anbrc 1265 . . 3 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (#‘𝐴) ∈ (ℤ𝑁))
63 vex 3234 . . . . 5 𝑎 ∈ V
64 ovex 6718 . . . . . 6 (1...𝑁) ∈ V
65 resiexg 7144 . . . . . 6 ((1...𝑁) ∈ V → ( I ↾ (1...𝑁)) ∈ V)
6664, 65ax-mp 5 . . . . 5 ( I ↾ (1...𝑁)) ∈ V
6763, 66unex 6998 . . . 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 6212 . . . . . 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 3838 . . . . . 6 (((𝑁 + 1)...(#‘𝐴)) ∩ (1...𝑁)) = ((1...𝑁) ∩ ((𝑁 + 1)...(#‘𝐴)))
7351nn0red 11390 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℝ)
7473ltp1d 10992 . . . . . . 7 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 < (𝑁 + 1))
75 fzdisj 12406 . . . . . . 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 2697 . . . . 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 6194 . . . . 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 1367 . . . 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 37634 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0𝑁 ≤ (#‘𝐴)) → (1...(#‘𝐴)) = ((1...𝑁) ∪ ((𝑁 + 1)...(#‘𝐴))))
8251, 55, 60, 81syl3anc 1366 . . . . . 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 3790 . . . . . 6 (((𝑁 + 1)...(#‘𝐴)) ∪ (1...𝑁)) = ((1...𝑁) ∪ ((𝑁 + 1)...(#‘𝐴)))
8482, 83syl6reqr 2704 . . . . 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 1086 . . . . . . 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 2697 . . . . 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 6168 . . . . 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 694 . . . 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 5446 . . . 4 ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ((𝑎 ↾ (1...𝑁)) ∪ (( I ↾ (1...𝑁)) ↾ (1...𝑁)))
92 dmres 5454 . . . . . . . 8 dom (𝑎 ↾ (1...𝑁)) = ((1...𝑁) ∩ dom 𝑎)
93 f1odm 6179 . . . . . . . . . . 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 3847 . . . . . . . . 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 2685 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ((1...𝑁) ∩ dom 𝑎) = ∅)
9792, 96syl5eq 2697 . . . . . . 7 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(#‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → dom (𝑎 ↾ (1...𝑁)) = ∅)
98 relres 5461 . . . . . . . 8 Rel (𝑎 ↾ (1...𝑁))
99 reldm0 5375 . . . . . . . 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 5465 . . . . . . 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 3801 . . . . 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 3790 . . . . . 6 (∅ ∪ ( I ↾ (1...𝑁))) = (( I ↾ (1...𝑁)) ∪ ∅)
106 un0 4000 . . . . . 6 (( I ↾ (1...𝑁)) ∪ ∅) = ( I ↾ (1...𝑁))
107105, 106eqtri 2673 . . . . 5 (∅ ∪ ( I ↾ (1...𝑁))) = ( I ↾ (1...𝑁))
108104, 107syl6eq 2701 . . . 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 2697 . . 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 6698 . . . . . 6 (𝑑 = (#‘𝐴) → (1...𝑑) = (1...(#‘𝐴)))
111 f1oeq2 6166 . . . . . 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 741 . . . 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 6165 . . . . 5 (𝑒 = (𝑎 ∪ ( I ↾ (1...𝑁))) → (𝑒:(1...(#‘𝐴))–1-1-onto𝐴 ↔ (𝑎 ∪ ( I ↾ (1...𝑁))):(1...(#‘𝐴))–1-1-onto𝐴))
115 reseq1 5422 . . . . . 6 (𝑒 = (𝑎 ∪ ( I ↾ (1...𝑁))) → (𝑒 ↾ (1...𝑁)) = ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)))
116115eqeq1d 2653 . . . . 5 (𝑒 = (𝑎 ∪ ( I ↾ (1...𝑁))) → ((𝑒 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)) ↔ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
117114, 116anbi12d 747 . . . 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 3355 . . 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 1370 . 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 1903 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 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wrex 2942  Vcvv 3231  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948   class class class wbr 4685   I cid 5052  dom cdm 5143  cres 5145  Rel wrel 5148  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  cen 7994  Fincfn 7997  cc 9972  cr 9973  1c1 9975   + caddc 9977   < clt 10112  cle 10113  0cn0 11330  cz 11415  cuz 11725  ...cfz 12364  #chash 13157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-hash 13158
This theorem is referenced by:  eldioph2  37642
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