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Theorem eldioph2lem2 39633
 Description: Lemma for eldioph2 39634. Construct necessary renaming function for one direction. (Contributed by Stefan O'Rear, 8-Oct-2014.)
Assertion
Ref Expression
eldioph2lem2 (((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) → ∃𝑐(𝑐:(1...𝐴)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
Distinct variable groups:   𝑁,𝑐   𝑆,𝑐   𝐴,𝑐

Proof of Theorem eldioph2lem2
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 simplr 768 . . . 4 (((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) → ¬ 𝑆 ∈ Fin)
2 fzfi 13335 . . . 4 (1...𝑁) ∈ Fin
3 difinf 8776 . . . 4 ((¬ 𝑆 ∈ Fin ∧ (1...𝑁) ∈ Fin) → ¬ (𝑆 ∖ (1...𝑁)) ∈ Fin)
41, 2, 3sylancl 589 . . 3 (((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) → ¬ (𝑆 ∖ (1...𝑁)) ∈ Fin)
5 fzfi 13335 . . . 4 (1...𝐴) ∈ Fin
6 diffi 8738 . . . 4 ((1...𝐴) ∈ Fin → ((1...𝐴) ∖ (1...𝑁)) ∈ Fin)
75, 6ax-mp 5 . . 3 ((1...𝐴) ∖ (1...𝑁)) ∈ Fin
8 isinffi 9409 . . 3 ((¬ (𝑆 ∖ (1...𝑁)) ∈ Fin ∧ ((1...𝐴) ∖ (1...𝑁)) ∈ Fin) → ∃𝑎 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)))
94, 7, 8sylancl 589 . 2 (((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) → ∃𝑎 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)))
10 f1f1orn 6608 . . . . . . . 8 (𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)) → 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1-onto→ran 𝑎)
1110adantl 485 . . . . . . 7 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1-onto→ran 𝑎)
12 f1oi 6634 . . . . . . . 8 ( I ↾ (1...𝑁)):(1...𝑁)–1-1-onto→(1...𝑁)
1312a1i 11 . . . . . . 7 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → ( I ↾ (1...𝑁)):(1...𝑁)–1-1-onto→(1...𝑁))
14 incom 4152 . . . . . . . . 9 (((1...𝐴) ∖ (1...𝑁)) ∩ (1...𝑁)) = ((1...𝑁) ∩ ((1...𝐴) ∖ (1...𝑁)))
15 disjdif 4393 . . . . . . . . 9 ((1...𝑁) ∩ ((1...𝐴) ∖ (1...𝑁))) = ∅
1614, 15eqtri 2845 . . . . . . . 8 (((1...𝐴) ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅
1716a1i 11 . . . . . . 7 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (((1...𝐴) ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅)
18 f1f 6556 . . . . . . . . . . . 12 (𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)) → 𝑎:((1...𝐴) ∖ (1...𝑁))⟶(𝑆 ∖ (1...𝑁)))
1918frnd 6501 . . . . . . . . . . 11 (𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)) → ran 𝑎 ⊆ (𝑆 ∖ (1...𝑁)))
2019adantl 485 . . . . . . . . . 10 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → ran 𝑎 ⊆ (𝑆 ∖ (1...𝑁)))
2120ssrind 4186 . . . . . . . . 9 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (ran 𝑎 ∩ (1...𝑁)) ⊆ ((𝑆 ∖ (1...𝑁)) ∩ (1...𝑁)))
22 incom 4152 . . . . . . . . . 10 ((𝑆 ∖ (1...𝑁)) ∩ (1...𝑁)) = ((1...𝑁) ∩ (𝑆 ∖ (1...𝑁)))
23 disjdif 4393 . . . . . . . . . 10 ((1...𝑁) ∩ (𝑆 ∖ (1...𝑁))) = ∅
2422, 23eqtri 2845 . . . . . . . . 9 ((𝑆 ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅
2521, 24sseqtrdi 3992 . . . . . . . 8 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (ran 𝑎 ∩ (1...𝑁)) ⊆ ∅)
26 ss0 4324 . . . . . . . 8 ((ran 𝑎 ∩ (1...𝑁)) ⊆ ∅ → (ran 𝑎 ∩ (1...𝑁)) = ∅)
2725, 26syl 17 . . . . . . 7 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (ran 𝑎 ∩ (1...𝑁)) = ∅)
28 f1oun 6616 . . . . . . 7 (((𝑎:((1...𝐴) ∖ (1...𝑁))–1-1-onto→ran 𝑎 ∧ ( I ↾ (1...𝑁)):(1...𝑁)–1-1-onto→(1...𝑁)) ∧ ((((1...𝐴) ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅ ∧ (ran 𝑎 ∩ (1...𝑁)) = ∅)) → (𝑎 ∪ ( I ↾ (1...𝑁))):(((1...𝐴) ∖ (1...𝑁)) ∪ (1...𝑁))–1-1-onto→(ran 𝑎 ∪ (1...𝑁)))
2911, 13, 17, 27, 28syl22anc 837 . . . . . 6 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (𝑎 ∪ ( I ↾ (1...𝑁))):(((1...𝐴) ∖ (1...𝑁)) ∪ (1...𝑁))–1-1-onto→(ran 𝑎 ∪ (1...𝑁)))
30 f1of1 6596 . . . . . 6 ((𝑎 ∪ ( I ↾ (1...𝑁))):(((1...𝐴) ∖ (1...𝑁)) ∪ (1...𝑁))–1-1-onto→(ran 𝑎 ∪ (1...𝑁)) → (𝑎 ∪ ( I ↾ (1...𝑁))):(((1...𝐴) ∖ (1...𝑁)) ∪ (1...𝑁))–1-1→(ran 𝑎 ∪ (1...𝑁)))
3129, 30syl 17 . . . . 5 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (𝑎 ∪ ( I ↾ (1...𝑁))):(((1...𝐴) ∖ (1...𝑁)) ∪ (1...𝑁))–1-1→(ran 𝑎 ∪ (1...𝑁)))
32 uncom 4104 . . . . . . 7 (((1...𝐴) ∖ (1...𝑁)) ∪ (1...𝑁)) = ((1...𝑁) ∪ ((1...𝐴) ∖ (1...𝑁)))
33 simplrr 777 . . . . . . . . 9 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → 𝐴 ∈ (ℤ𝑁))
34 fzss2 12942 . . . . . . . . 9 (𝐴 ∈ (ℤ𝑁) → (1...𝑁) ⊆ (1...𝐴))
3533, 34syl 17 . . . . . . . 8 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (1...𝑁) ⊆ (1...𝐴))
36 undif 4402 . . . . . . . 8 ((1...𝑁) ⊆ (1...𝐴) ↔ ((1...𝑁) ∪ ((1...𝐴) ∖ (1...𝑁))) = (1...𝐴))
3735, 36sylib 221 . . . . . . 7 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → ((1...𝑁) ∪ ((1...𝐴) ∖ (1...𝑁))) = (1...𝐴))
3832, 37syl5eq 2869 . . . . . 6 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (((1...𝐴) ∖ (1...𝑁)) ∪ (1...𝑁)) = (1...𝐴))
39 f1eq2 6552 . . . . . 6 ((((1...𝐴) ∖ (1...𝑁)) ∪ (1...𝑁)) = (1...𝐴) → ((𝑎 ∪ ( I ↾ (1...𝑁))):(((1...𝐴) ∖ (1...𝑁)) ∪ (1...𝑁))–1-1→(ran 𝑎 ∪ (1...𝑁)) ↔ (𝑎 ∪ ( I ↾ (1...𝑁))):(1...𝐴)–1-1→(ran 𝑎 ∪ (1...𝑁))))
4038, 39syl 17 . . . . 5 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → ((𝑎 ∪ ( I ↾ (1...𝑁))):(((1...𝐴) ∖ (1...𝑁)) ∪ (1...𝑁))–1-1→(ran 𝑎 ∪ (1...𝑁)) ↔ (𝑎 ∪ ( I ↾ (1...𝑁))):(1...𝐴)–1-1→(ran 𝑎 ∪ (1...𝑁))))
4131, 40mpbid 235 . . . 4 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (𝑎 ∪ ( I ↾ (1...𝑁))):(1...𝐴)–1-1→(ran 𝑎 ∪ (1...𝑁)))
4219difss2d 4086 . . . . . 6 (𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)) → ran 𝑎𝑆)
4342adantl 485 . . . . 5 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → ran 𝑎𝑆)
44 simplrl 776 . . . . 5 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (1...𝑁) ⊆ 𝑆)
4543, 44unssd 4137 . . . 4 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (ran 𝑎 ∪ (1...𝑁)) ⊆ 𝑆)
46 f1ss 6562 . . . 4 (((𝑎 ∪ ( I ↾ (1...𝑁))):(1...𝐴)–1-1→(ran 𝑎 ∪ (1...𝑁)) ∧ (ran 𝑎 ∪ (1...𝑁)) ⊆ 𝑆) → (𝑎 ∪ ( I ↾ (1...𝑁))):(1...𝐴)–1-1𝑆)
4741, 45, 46syl2anc 587 . . 3 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (𝑎 ∪ ( I ↾ (1...𝑁))):(1...𝐴)–1-1𝑆)
48 resundir 5846 . . . 4 ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ((𝑎 ↾ (1...𝑁)) ∪ (( I ↾ (1...𝑁)) ↾ (1...𝑁)))
49 dmres 5853 . . . . . . . 8 dom (𝑎 ↾ (1...𝑁)) = ((1...𝑁) ∩ dom 𝑎)
50 incom 4152 . . . . . . . . 9 ((1...𝑁) ∩ dom 𝑎) = (dom 𝑎 ∩ (1...𝑁))
51 f1dm 6560 . . . . . . . . . . . 12 (𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)) → dom 𝑎 = ((1...𝐴) ∖ (1...𝑁)))
5251adantl 485 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → dom 𝑎 = ((1...𝐴) ∖ (1...𝑁)))
5352ineq1d 4162 . . . . . . . . . 10 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (dom 𝑎 ∩ (1...𝑁)) = (((1...𝐴) ∖ (1...𝑁)) ∩ (1...𝑁)))
5453, 16syl6eq 2873 . . . . . . . . 9 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (dom 𝑎 ∩ (1...𝑁)) = ∅)
5550, 54syl5eq 2869 . . . . . . . 8 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → ((1...𝑁) ∩ dom 𝑎) = ∅)
5649, 55syl5eq 2869 . . . . . . 7 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → dom (𝑎 ↾ (1...𝑁)) = ∅)
57 relres 5860 . . . . . . . 8 Rel (𝑎 ↾ (1...𝑁))
58 reldm0 5775 . . . . . . . 8 (Rel (𝑎 ↾ (1...𝑁)) → ((𝑎 ↾ (1...𝑁)) = ∅ ↔ dom (𝑎 ↾ (1...𝑁)) = ∅))
5957, 58ax-mp 5 . . . . . . 7 ((𝑎 ↾ (1...𝑁)) = ∅ ↔ dom (𝑎 ↾ (1...𝑁)) = ∅)
6056, 59sylibr 237 . . . . . 6 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (𝑎 ↾ (1...𝑁)) = ∅)
61 residm 5864 . . . . . . 7 (( I ↾ (1...𝑁)) ↾ (1...𝑁)) = ( I ↾ (1...𝑁))
6261a1i 11 . . . . . 6 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (( I ↾ (1...𝑁)) ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))
6360, 62uneq12d 4115 . . . . 5 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → ((𝑎 ↾ (1...𝑁)) ∪ (( I ↾ (1...𝑁)) ↾ (1...𝑁))) = (∅ ∪ ( I ↾ (1...𝑁))))
64 uncom 4104 . . . . . 6 (∅ ∪ ( I ↾ (1...𝑁))) = (( I ↾ (1...𝑁)) ∪ ∅)
65 un0 4316 . . . . . 6 (( I ↾ (1...𝑁)) ∪ ∅) = ( I ↾ (1...𝑁))
6664, 65eqtri 2845 . . . . 5 (∅ ∪ ( I ↾ (1...𝑁))) = ( I ↾ (1...𝑁))
6763, 66syl6eq 2873 . . . 4 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → ((𝑎 ↾ (1...𝑁)) ∪ (( I ↾ (1...𝑁)) ↾ (1...𝑁))) = ( I ↾ (1...𝑁)))
6848, 67syl5eq 2869 . . 3 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))
69 vex 3472 . . . . 5 𝑎 ∈ V
70 ovex 7173 . . . . . 6 (1...𝑁) ∈ V
71 resiexg 7605 . . . . . 6 ((1...𝑁) ∈ V → ( I ↾ (1...𝑁)) ∈ V)
7270, 71ax-mp 5 . . . . 5 ( I ↾ (1...𝑁)) ∈ V
7369, 72unex 7454 . . . 4 (𝑎 ∪ ( I ↾ (1...𝑁))) ∈ V
74 f1eq1 6551 . . . . 5 (𝑐 = (𝑎 ∪ ( I ↾ (1...𝑁))) → (𝑐:(1...𝐴)–1-1𝑆 ↔ (𝑎 ∪ ( I ↾ (1...𝑁))):(1...𝐴)–1-1𝑆))
75 reseq1 5825 . . . . . 6 (𝑐 = (𝑎 ∪ ( I ↾ (1...𝑁))) → (𝑐 ↾ (1...𝑁)) = ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)))
7675eqeq1d 2824 . . . . 5 (𝑐 = (𝑎 ∪ ( I ↾ (1...𝑁))) → ((𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)) ↔ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
7774, 76anbi12d 633 . . . 4 (𝑐 = (𝑎 ∪ ( I ↾ (1...𝑁))) → ((𝑐:(1...𝐴)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) ↔ ((𝑎 ∪ ( I ↾ (1...𝑁))):(1...𝐴)–1-1𝑆 ∧ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))))
7873, 77spcev 3582 . . 3 (((𝑎 ∪ ( I ↾ (1...𝑁))):(1...𝐴)–1-1𝑆 ∧ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) → ∃𝑐(𝑐:(1...𝐴)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
7947, 68, 78syl2anc 587 . 2 ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → ∃𝑐(𝑐:(1...𝐴)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
809, 79exlimddv 1936 1 (((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆𝐴 ∈ (ℤ𝑁))) → ∃𝑐(𝑐:(1...𝐴)–1-1𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2114  Vcvv 3469   ∖ cdif 3905   ∪ cun 3906   ∩ cin 3907   ⊆ wss 3908  ∅c0 4265   I cid 5436  dom cdm 5532  ran crn 5533   ↾ cres 5534  Rel wrel 5537  –1-1→wf1 6331  –1-1-onto→wf1o 6333  ‘cfv 6334  (class class class)co 7140  Fincfn 8496  1c1 10527  ℕ0cn0 11885  ℤ≥cuz 12231  ...cfz 12885 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-card 9356  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886 This theorem is referenced by:  eldioph2b  39635
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