eldioph2lem2 - Mathbox for Stefan O'Rear

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Theorem eldioph2lem2 36342

Description: Lemma for eldioph2 36343. 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) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) → ∃𝑐(𝑐:(1...𝐴)–1-1→𝑆 ∧ (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.

Step	Hyp	Ref	Expression
1		simplr 788	. . . 4 ⊢ (((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) → ¬ 𝑆 ∈ Fin)
2		fzfi 12633	. . . 4 ⊢ (1...𝑁) ∈ Fin
3		difinf 8115	. . . 4 ⊢ ((¬ 𝑆 ∈ Fin ∧ (1...𝑁) ∈ Fin) → ¬ (𝑆 ∖ (1...𝑁)) ∈ Fin)
4	1, 2, 3	sylancl 693	. . 3 ⊢ (((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) → ¬ (𝑆 ∖ (1...𝑁)) ∈ Fin)
5		fzfi 12633	. . . 4 ⊢ (1...𝐴) ∈ Fin
6		diffi 8077	. . . 4 ⊢ ((1...𝐴) ∈ Fin → ((1...𝐴) ∖ (1...𝑁)) ∈ Fin)
7	5, 6	ax-mp 5	. . 3 ⊢ ((1...𝐴) ∖ (1...𝑁)) ∈ Fin
8		isinffi 8701	. . 3 ⊢ ((¬ (𝑆 ∖ (1...𝑁)) ∈ Fin ∧ ((1...𝐴) ∖ (1...𝑁)) ∈ Fin) → ∃𝑎 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)))
9	4, 7, 8	sylancl 693	. 2 ⊢ (((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) → ∃𝑎 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)))
10		f1f1orn 6061	. . . . . . . 8 ⊢ (𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)) → 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1-onto→ran 𝑎)
11	10	adantl 481	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1-onto→ran 𝑎)
12		f1oi 6086	. . . . . . . 8 ⊢ ( I ↾ (1...𝑁)):(1...𝑁)–1-1-onto→(1...𝑁)
13	12	a1i 11	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → ( I ↾ (1...𝑁)):(1...𝑁)–1-1-onto→(1...𝑁))
14		incom 3767	. . . . . . . . 9 ⊢ (((1...𝐴) ∖ (1...𝑁)) ∩ (1...𝑁)) = ((1...𝑁) ∩ ((1...𝐴) ∖ (1...𝑁)))
15		disjdif 3992	. . . . . . . . 9 ⊢ ((1...𝑁) ∩ ((1...𝐴) ∖ (1...𝑁))) = ∅
16	14, 15	eqtri 2632	. . . . . . . 8 ⊢ (((1...𝐴) ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅
17	16	a1i 11	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (((1...𝐴) ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅)
18		f1f 6014	. . . . . . . . . . . 12 ⊢ (𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)) → 𝑎:((1...𝐴) ∖ (1...𝑁))⟶(𝑆 ∖ (1...𝑁)))
19		frn 5966	. . . . . . . . . . . 12 ⊢ (𝑎:((1...𝐴) ∖ (1...𝑁))⟶(𝑆 ∖ (1...𝑁)) → ran 𝑎 ⊆ (𝑆 ∖ (1...𝑁)))
20	18, 19	syl 17	. . . . . . . . . . 11 ⊢ (𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)) → ran 𝑎 ⊆ (𝑆 ∖ (1...𝑁)))
21	20	adantl 481	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → ran 𝑎 ⊆ (𝑆 ∖ (1...𝑁)))
22		ssrin 3800	. . . . . . . . . 10 ⊢ (ran 𝑎 ⊆ (𝑆 ∖ (1...𝑁)) → (ran 𝑎 ∩ (1...𝑁)) ⊆ ((𝑆 ∖ (1...𝑁)) ∩ (1...𝑁)))
23	21, 22	syl 17	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (ran 𝑎 ∩ (1...𝑁)) ⊆ ((𝑆 ∖ (1...𝑁)) ∩ (1...𝑁)))
24		incom 3767	. . . . . . . . . 10 ⊢ ((𝑆 ∖ (1...𝑁)) ∩ (1...𝑁)) = ((1...𝑁) ∩ (𝑆 ∖ (1...𝑁)))
25		disjdif 3992	. . . . . . . . . 10 ⊢ ((1...𝑁) ∩ (𝑆 ∖ (1...𝑁))) = ∅
26	24, 25	eqtri 2632	. . . . . . . . 9 ⊢ ((𝑆 ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅
27	23, 26	syl6sseq 3614	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (ran 𝑎 ∩ (1...𝑁)) ⊆ ∅)
28		ss0 3926	. . . . . . . 8 ⊢ ((ran 𝑎 ∩ (1...𝑁)) ⊆ ∅ → (ran 𝑎 ∩ (1...𝑁)) = ∅)
29	27, 28	syl 17	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (ran 𝑎 ∩ (1...𝑁)) = ∅)
30		f1oun 6069	. . . . . . 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)))
31	11, 13, 17, 29, 30	syl22anc 1319	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 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)))
32		f1of1 6049	. . . . . 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)))
33	31, 32	syl 17	. . . . 5 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 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)))
34		uncom 3719	. . . . . . 7 ⊢ (((1...𝐴) ∖ (1...𝑁)) ∪ (1...𝑁)) = ((1...𝑁) ∪ ((1...𝐴) ∖ (1...𝑁)))
35		simplrr 797	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → 𝐴 ∈ (ℤ_≥‘𝑁))
36		fzss2 12252	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℤ_≥‘𝑁) → (1...𝑁) ⊆ (1...𝐴))
37	35, 36	syl 17	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (1...𝑁) ⊆ (1...𝐴))
38		undif 4001	. . . . . . . 8 ⊢ ((1...𝑁) ⊆ (1...𝐴) ↔ ((1...𝑁) ∪ ((1...𝐴) ∖ (1...𝑁))) = (1...𝐴))
39	37, 38	sylib 207	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → ((1...𝑁) ∪ ((1...𝐴) ∖ (1...𝑁))) = (1...𝐴))
40	34, 39	syl5eq 2656	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (((1...𝐴) ∖ (1...𝑁)) ∪ (1...𝑁)) = (1...𝐴))
41		f1eq2 6010	. . . . . 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))))
42	40, 41	syl 17	. . . . 5 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 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))))
43	33, 42	mpbid 221	. . . 4 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (𝑎 ∪ ( I ↾ (1...𝑁))):(1...𝐴)–1-1→(ran 𝑎 ∪ (1...𝑁)))
44	20	difss2d 3702	. . . . . 6 ⊢ (𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)) → ran 𝑎 ⊆ 𝑆)
45	44	adantl 481	. . . . 5 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → ran 𝑎 ⊆ 𝑆)
46		simplrl 796	. . . . 5 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (1...𝑁) ⊆ 𝑆)
47	45, 46	unssd 3751	. . . 4 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (ran 𝑎 ∪ (1...𝑁)) ⊆ 𝑆)
48		f1ss 6019	. . . 4 ⊢ (((𝑎 ∪ ( I ↾ (1...𝑁))):(1...𝐴)–1-1→(ran 𝑎 ∪ (1...𝑁)) ∧ (ran 𝑎 ∪ (1...𝑁)) ⊆ 𝑆) → (𝑎 ∪ ( I ↾ (1...𝑁))):(1...𝐴)–1-1→𝑆)
49	43, 47, 48	syl2anc 691	. . 3 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (𝑎 ∪ ( I ↾ (1...𝑁))):(1...𝐴)–1-1→𝑆)
50		resundir 5331	. . . 4 ⊢ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ((𝑎 ↾ (1...𝑁)) ∪ (( I ↾ (1...𝑁)) ↾ (1...𝑁)))
51		dmres 5339	. . . . . . . 8 ⊢ dom (𝑎 ↾ (1...𝑁)) = ((1...𝑁) ∩ dom 𝑎)
52		incom 3767	. . . . . . . . 9 ⊢ ((1...𝑁) ∩ dom 𝑎) = (dom 𝑎 ∩ (1...𝑁))
53		f1dm 6018	. . . . . . . . . . . 12 ⊢ (𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)) → dom 𝑎 = ((1...𝐴) ∖ (1...𝑁)))
54	53	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → dom 𝑎 = ((1...𝐴) ∖ (1...𝑁)))
55	54	ineq1d 3775	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (dom 𝑎 ∩ (1...𝑁)) = (((1...𝐴) ∖ (1...𝑁)) ∩ (1...𝑁)))
56	55, 16	syl6eq 2660	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (dom 𝑎 ∩ (1...𝑁)) = ∅)
57	52, 56	syl5eq 2656	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → ((1...𝑁) ∩ dom 𝑎) = ∅)
58	51, 57	syl5eq 2656	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → dom (𝑎 ↾ (1...𝑁)) = ∅)
59		relres 5346	. . . . . . . 8 ⊢ Rel (𝑎 ↾ (1...𝑁))
60		reldm0 5264	. . . . . . . 8 ⊢ (Rel (𝑎 ↾ (1...𝑁)) → ((𝑎 ↾ (1...𝑁)) = ∅ ↔ dom (𝑎 ↾ (1...𝑁)) = ∅))
61	59, 60	ax-mp 5	. . . . . . 7 ⊢ ((𝑎 ↾ (1...𝑁)) = ∅ ↔ dom (𝑎 ↾ (1...𝑁)) = ∅)
62	58, 61	sylibr 223	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (𝑎 ↾ (1...𝑁)) = ∅)
63		residm 5350	. . . . . . 7 ⊢ (( I ↾ (1...𝑁)) ↾ (1...𝑁)) = ( I ↾ (1...𝑁))
64	63	a1i 11	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (( I ↾ (1...𝑁)) ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))
65	62, 64	uneq12d 3730	. . . . 5 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → ((𝑎 ↾ (1...𝑁)) ∪ (( I ↾ (1...𝑁)) ↾ (1...𝑁))) = (∅ ∪ ( I ↾ (1...𝑁))))
66		uncom 3719	. . . . . 6 ⊢ (∅ ∪ ( I ↾ (1...𝑁))) = (( I ↾ (1...𝑁)) ∪ ∅)
67		un0 3919	. . . . . 6 ⊢ (( I ↾ (1...𝑁)) ∪ ∅) = ( I ↾ (1...𝑁))
68	66, 67	eqtri 2632	. . . . 5 ⊢ (∅ ∪ ( I ↾ (1...𝑁))) = ( I ↾ (1...𝑁))
69	65, 68	syl6eq 2660	. . . 4 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → ((𝑎 ↾ (1...𝑁)) ∪ (( I ↾ (1...𝑁)) ↾ (1...𝑁))) = ( I ↾ (1...𝑁)))
70	50, 69	syl5eq 2656	. . 3 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))
71		vex 3176	. . . . 5 ⊢ 𝑎 ∈ V
72		ovex 6577	. . . . . 6 ⊢ (1...𝑁) ∈ V
73		resiexg 6994	. . . . . 6 ⊢ ((1...𝑁) ∈ V → ( I ↾ (1...𝑁)) ∈ V)
74	72, 73	ax-mp 5	. . . . 5 ⊢ ( I ↾ (1...𝑁)) ∈ V
75	71, 74	unex 6854	. . . 4 ⊢ (𝑎 ∪ ( I ↾ (1...𝑁))) ∈ V
76		f1eq1 6009	. . . . 5 ⊢ (𝑐 = (𝑎 ∪ ( I ↾ (1...𝑁))) → (𝑐:(1...𝐴)–1-1→𝑆 ↔ (𝑎 ∪ ( I ↾ (1...𝑁))):(1...𝐴)–1-1→𝑆))
77		reseq1 5311	. . . . . 6 ⊢ (𝑐 = (𝑎 ∪ ( I ↾ (1...𝑁))) → (𝑐 ↾ (1...𝑁)) = ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)))
78	77	eqeq1d 2612	. . . . 5 ⊢ (𝑐 = (𝑎 ∪ ( I ↾ (1...𝑁))) → ((𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)) ↔ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
79	76, 78	anbi12d 743	. . . 4 ⊢ (𝑐 = (𝑎 ∪ ( I ↾ (1...𝑁))) → ((𝑐:(1...𝐴)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) ↔ ((𝑎 ∪ ( I ↾ (1...𝑁))):(1...𝐴)–1-1→𝑆 ∧ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))))
80	75, 79	spcev 3273	. . 3 ⊢ (((𝑎 ∪ ( I ↾ (1...𝑁))):(1...𝐴)–1-1→𝑆 ∧ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) → ∃𝑐(𝑐:(1...𝐴)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
81	49, 70, 80	syl2anc 691	. 2 ⊢ ((((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → ∃𝑐(𝑐:(1...𝐴)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
82	9, 81	exlimddv 1850	1 ⊢ (((𝑁 ∈ ℕ₀ ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ_≥‘𝑁))) → ∃𝑐(𝑐:(1...𝐴)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 I cid 4948 dom cdm 5038 ran crn 5039 ↾ cres 5040 Rel wrel 5043 ⟶wf 5800 –1-1→wf1 5801 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 1c1 9816 ℕ₀cn0 11169 ℤ_≥cuz 11563 ...cfz 12197

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-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-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-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

This theorem is referenced by: eldioph2b 36344

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