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Theorem eldioph2lem1 37822
Description: Lemma for eldioph2 37824. 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 11564 . . . . . . . . . 10 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
213ad2ant1 1156 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℝ)
32recnd 10350 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℂ)
4 ax-1cn 10276 . . . . . . . 8 1 ∈ ℂ
5 addcom 10504 . . . . . . . 8 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 + 1) = (1 + 𝑁))
63, 4, 5sylancl 576 . . . . . . 7 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (𝑁 + 1) = (1 + 𝑁))
7 diffi 8428 . . . . . . . . . 10 (𝐴 ∈ Fin → (𝐴 ∖ (1...𝑁)) ∈ Fin)
873ad2ant2 1157 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (𝐴 ∖ (1...𝑁)) ∈ Fin)
9 fzfid 12992 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...𝑁) ∈ Fin)
10 incom 4001 . . . . . . . . . . 11 ((𝐴 ∖ (1...𝑁)) ∩ (1...𝑁)) = ((1...𝑁) ∩ (𝐴 ∖ (1...𝑁)))
11 disjdif 4233 . . . . . . . . . . 11 ((1...𝑁) ∩ (𝐴 ∖ (1...𝑁))) = ∅
1210, 11eqtri 2827 . . . . . . . . . 10 ((𝐴 ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅
1312a1i 11 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝐴 ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅)
14 hashun 13385 . . . . . . . . 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 1483 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁))) = ((♯‘(𝐴 ∖ (1...𝑁))) + (♯‘(1...𝑁))))
16 uncom 3953 . . . . . . . . . 10 ((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁)) = ((1...𝑁) ∪ (𝐴 ∖ (1...𝑁)))
17 simp3 1161 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...𝑁) ⊆ 𝐴)
18 undif 4242 . . . . . . . . . . 11 ((1...𝑁) ⊆ 𝐴 ↔ ((1...𝑁) ∪ (𝐴 ∖ (1...𝑁))) = 𝐴)
1917, 18sylib 209 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((1...𝑁) ∪ (𝐴 ∖ (1...𝑁))) = 𝐴)
2016, 19syl5eq 2851 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁)) = 𝐴)
2120fveq2d 6409 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((𝐴 ∖ (1...𝑁)) ∪ (1...𝑁))) = (♯‘𝐴))
22 hashfz1 13350 . . . . . . . . . 10 (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
23223ad2ant1 1156 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘(1...𝑁)) = 𝑁)
2423oveq2d 6887 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((♯‘(𝐴 ∖ (1...𝑁))) + (♯‘(1...𝑁))) = ((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))
2515, 21, 243eqtr3d 2847 . . . . . . 7 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘𝐴) = ((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))
266, 25oveq12d 6889 . . . . . 6 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝑁 + 1)...(♯‘𝐴)) = ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)))
2726fveq2d 6409 . . . . 5 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((𝑁 + 1)...(♯‘𝐴))) = (♯‘((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))))
28 1zzd 11670 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 1 ∈ ℤ)
29 hashcl 13361 . . . . . . . . . 10 ((𝐴 ∖ (1...𝑁)) ∈ Fin → (♯‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0)
308, 29syl 17 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0)
3130nn0zd 11742 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘(𝐴 ∖ (1...𝑁))) ∈ ℤ)
32 nn0z 11662 . . . . . . . . 9 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
33323ad2ant1 1156 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ∈ ℤ)
34 fzen 12577 . . . . . . . 8 ((1 ∈ ℤ ∧ (♯‘(𝐴 ∖ (1...𝑁))) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (1...(♯‘(𝐴 ∖ (1...𝑁)))) ≈ ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)))
3528, 31, 33, 34syl3anc 1483 . . . . . . 7 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (1...(♯‘(𝐴 ∖ (1...𝑁)))) ≈ ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)))
3635ensymd 8240 . . . . . 6 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)) ≈ (1...(♯‘(𝐴 ∖ (1...𝑁)))))
37 fzfi 12991 . . . . . . 7 ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)) ∈ Fin
38 fzfi 12991 . . . . . . 7 (1...(♯‘(𝐴 ∖ (1...𝑁)))) ∈ Fin
39 hashen 13351 . . . . . . 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 675 . . . . . 6 ((♯‘((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))) = (♯‘(1...(♯‘(𝐴 ∖ (1...𝑁))))) ↔ ((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁)) ≈ (1...(♯‘(𝐴 ∖ (1...𝑁)))))
4136, 40sylibr 225 . . . . 5 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((1 + 𝑁)...((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))) = (♯‘(1...(♯‘(𝐴 ∖ (1...𝑁))))))
42 hashfz1 13350 . . . . . 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 2843 . . . 4 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → (♯‘((𝑁 + 1)...(♯‘𝐴))) = (♯‘(𝐴 ∖ (1...𝑁))))
45 fzfi 12991 . . . . 5 ((𝑁 + 1)...(♯‘𝐴)) ∈ Fin
46 hashen 13351 . . . . 5 ((((𝑁 + 1)...(♯‘𝐴)) ∈ Fin ∧ (𝐴 ∖ (1...𝑁)) ∈ Fin) → ((♯‘((𝑁 + 1)...(♯‘𝐴))) = (♯‘(𝐴 ∖ (1...𝑁))) ↔ ((𝑁 + 1)...(♯‘𝐴)) ≈ (𝐴 ∖ (1...𝑁))))
4745, 8, 46sylancr 577 . . . 4 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((♯‘((𝑁 + 1)...(♯‘𝐴))) = (♯‘(𝐴 ∖ (1...𝑁))) ↔ ((𝑁 + 1)...(♯‘𝐴)) ≈ (𝐴 ∖ (1...𝑁))))
4844, 47mpbid 223 . . 3 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ((𝑁 + 1)...(♯‘𝐴)) ≈ (𝐴 ∖ (1...𝑁)))
49 bren 8198 . . 3 (((𝑁 + 1)...(♯‘𝐴)) ≈ (𝐴 ∖ (1...𝑁)) ↔ ∃𝑎 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁)))
5048, 49sylib 209 . 2 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → ∃𝑎 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁)))
51 simpl1 1235 . . . . 5 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℕ0)
5251nn0zd 11742 . . . 4 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℤ)
53 simpl2 1237 . . . . . 6 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝐴 ∈ Fin)
54 hashcl 13361 . . . . . 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 11742 . . . 4 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (♯‘𝐴) ∈ ℤ)
57 nn0addge2 11602 . . . . . . 7 ((𝑁 ∈ ℝ ∧ (♯‘(𝐴 ∖ (1...𝑁))) ∈ ℕ0) → 𝑁 ≤ ((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))
582, 30, 57syl2anc 575 . . . . . 6 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ≤ ((♯‘(𝐴 ∖ (1...𝑁))) + 𝑁))
5958, 25breqtrrd 4868 . . . . 5 ((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) → 𝑁 ≤ (♯‘𝐴))
6059adantr 468 . . . 4 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ≤ (♯‘𝐴))
61 eluz2 11906 . . . 4 ((♯‘𝐴) ∈ (ℤ𝑁) ↔ (𝑁 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑁 ≤ (♯‘𝐴)))
6252, 56, 60, 61syl3anbrc 1436 . . 3 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (♯‘𝐴) ∈ (ℤ𝑁))
63 vex 3393 . . . . 5 𝑎 ∈ V
64 ovex 6903 . . . . . 6 (1...𝑁) ∈ V
65 resiexg 7329 . . . . . 6 ((1...𝑁) ∈ V → ( I ↾ (1...𝑁)) ∈ V)
6664, 65ax-mp 5 . . . . 5 ( I ↾ (1...𝑁)) ∈ V
6763, 66unex 7183 . . . 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 473 . . . . 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 6387 . . . . . 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 4001 . . . . . 6 (((𝑁 + 1)...(♯‘𝐴)) ∩ (1...𝑁)) = ((1...𝑁) ∩ ((𝑁 + 1)...(♯‘𝐴)))
7351nn0red 11614 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 ∈ ℝ)
7473ltp1d 11236 . . . . . . 7 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → 𝑁 < (𝑁 + 1))
75 fzdisj 12587 . . . . . . 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 2851 . . . . 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 6369 . . . . 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 858 . . . 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 37816 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0𝑁 ≤ (♯‘𝐴)) → (1...(♯‘𝐴)) = ((1...𝑁) ∪ ((𝑁 + 1)...(♯‘𝐴))))
8251, 55, 60, 81syl3anc 1483 . . . . . 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 3953 . . . . . 6 (((𝑁 + 1)...(♯‘𝐴)) ∪ (1...𝑁)) = ((1...𝑁) ∪ ((𝑁 + 1)...(♯‘𝐴)))
8482, 83syl6reqr 2858 . . . . 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 1239 . . . . . . 7 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (1...𝑁) ⊆ 𝐴)
8685, 18sylib 209 . . . . . 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 2851 . . . . 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 6343 . . . . 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 575 . . . 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 223 . . 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 5612 . . . 4 ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ((𝑎 ↾ (1...𝑁)) ∪ (( I ↾ (1...𝑁)) ↾ (1...𝑁)))
92 dmres 5619 . . . . . . . 8 dom (𝑎 ↾ (1...𝑁)) = ((1...𝑁) ∩ dom 𝑎)
93 f1odm 6354 . . . . . . . . . . 11 (𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁)) → dom 𝑎 = ((𝑁 + 1)...(♯‘𝐴)))
9493adantl 469 . . . . . . . . . 10 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → dom 𝑎 = ((𝑁 + 1)...(♯‘𝐴)))
9594ineq2d 4010 . . . . . . . . 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 2839 . . . . . . . 8 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → ((1...𝑁) ∩ dom 𝑎) = ∅)
9792, 96syl5eq 2851 . . . . . . 7 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → dom (𝑎 ↾ (1...𝑁)) = ∅)
98 relres 5626 . . . . . . . 8 Rel (𝑎 ↾ (1...𝑁))
99 reldm0 5541 . . . . . . . 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 225 . . . . . 6 (((𝑁 ∈ ℕ0𝐴 ∈ Fin ∧ (1...𝑁) ⊆ 𝐴) ∧ 𝑎:((𝑁 + 1)...(♯‘𝐴))–1-1-onto→(𝐴 ∖ (1...𝑁))) → (𝑎 ↾ (1...𝑁)) = ∅)
102 residm 5630 . . . . . . 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 3964 . . . . 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 3953 . . . . . 6 (∅ ∪ ( I ↾ (1...𝑁))) = (( I ↾ (1...𝑁)) ∪ ∅)
106 un0 4162 . . . . . 6 (( I ↾ (1...𝑁)) ∪ ∅) = ( I ↾ (1...𝑁))
107105, 106eqtri 2827 . . . . 5 (∅ ∪ ( I ↾ (1...𝑁))) = ( I ↾ (1...𝑁))
108104, 107syl6eq 2855 . . . 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 2851 . . 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 6879 . . . . . 6 (𝑑 = (♯‘𝐴) → (1...𝑑) = (1...(♯‘𝐴)))
111 f1oeq2 6341 . . . . . 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 617 . . . 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 6340 . . . . 5 (𝑒 = (𝑎 ∪ ( I ↾ (1...𝑁))) → (𝑒:(1...(♯‘𝐴))–1-1-onto𝐴 ↔ (𝑎 ∪ ( I ↾ (1...𝑁))):(1...(♯‘𝐴))–1-1-onto𝐴))
115 reseq1 5588 . . . . . 6 (𝑒 = (𝑎 ∪ ( I ↾ (1...𝑁))) → (𝑒 ↾ (1...𝑁)) = ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)))
116115eqeq1d 2807 . . . . 5 (𝑒 = (𝑎 ∪ ( I ↾ (1...𝑁))) → ((𝑒 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)) ↔ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
117114, 116anbi12d 618 . . . 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 3516 . . 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 1486 . 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 2028 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 197  wa 384  w3a 1100   = wceq 1637  wex 1859  wcel 2158  wrex 3096  Vcvv 3390  cdif 3763  cun 3764  cin 3765  wss 3766  c0 4113   class class class wbr 4840   I cid 5215  dom cdm 5308  cres 5310  Rel wrel 5313  1-1-ontowf1o 6097  cfv 6098  (class class class)co 6871  cen 8186  Fincfn 8189  cc 10216  cr 10217  1c1 10219   + caddc 10221   < clt 10356  cle 10357  0cn0 11555  cz 11639  cuz 11900  ...cfz 12545  chash 13333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-rep 4960  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176  ax-cnex 10274  ax-resscn 10275  ax-1cn 10276  ax-icn 10277  ax-addcl 10278  ax-addrcl 10279  ax-mulcl 10280  ax-mulrcl 10281  ax-mulcom 10282  ax-addass 10283  ax-mulass 10284  ax-distr 10285  ax-i2m1 10286  ax-1ne0 10287  ax-1rid 10288  ax-rnegex 10289  ax-rrecex 10290  ax-cnre 10291  ax-pre-lttri 10292  ax-pre-lttrn 10293  ax-pre-ltadd 10294  ax-pre-mulgt0 10295
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-nel 3081  df-ral 3100  df-rex 3101  df-reu 3102  df-rmo 3103  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-pss 3782  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-tp 4372  df-op 4374  df-uni 4627  df-int 4666  df-iun 4710  df-br 4841  df-opab 4903  df-mpt 4920  df-tr 4943  df-id 5216  df-eprel 5221  df-po 5229  df-so 5230  df-fr 5267  df-we 5269  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-pred 5890  df-ord 5936  df-on 5937  df-lim 5938  df-suc 5939  df-iota 6061  df-fun 6100  df-fn 6101  df-f 6102  df-f1 6103  df-fo 6104  df-f1o 6105  df-fv 6106  df-riota 6832  df-ov 6874  df-oprab 6875  df-mpt2 6876  df-om 7293  df-1st 7395  df-2nd 7396  df-wrecs 7639  df-recs 7701  df-rdg 7739  df-1o 7793  df-oadd 7797  df-er 7976  df-en 8190  df-dom 8191  df-sdom 8192  df-fin 8193  df-card 9045  df-cda 9272  df-pnf 10358  df-mnf 10359  df-xr 10360  df-ltxr 10361  df-le 10362  df-sub 10550  df-neg 10551  df-nn 11303  df-n0 11556  df-z 11640  df-uz 11901  df-fz 12546  df-hash 13334
This theorem is referenced by:  eldioph2  37824
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