Theorem eldioph2lem2 42754

Description: Lemma for eldioph2 42755. 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 768	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ≥‘𝑁))) → ¬ 𝑆 ∈ Fin)
2		fzfi 13879	. . . 4 ⊢ (1...𝑁) ∈ Fin
3		difinf 9200	. . . 4 ⊢ ((¬ 𝑆 ∈ Fin ∧ (1...𝑁) ∈ Fin) → ¬ (𝑆 ∖ (1...𝑁)) ∈ Fin)
4	1, 2, 3	sylancl 586	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ≥‘𝑁))) → ¬ (𝑆 ∖ (1...𝑁)) ∈ Fin)
5		fzfi 13879	. . . 4 ⊢ (1...𝐴) ∈ Fin
6		diffi 9089	. . . 4 ⊢ ((1...𝐴) ∈ Fin → ((1...𝐴) ∖ (1...𝑁)) ∈ Fin)
7	5, 6	ax-mp 5	. . 3 ⊢ ((1...𝐴) ∖ (1...𝑁)) ∈ Fin
8		isinffi 9888	. . 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 586	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ≥‘𝑁))) → ∃𝑎 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)))
10		f1f1orn 6775	. . . . . . . 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 6802	. . . . . . . 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		disjdifr 4424	. . . . . . . 8 ⊢ (((1...𝐴) ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅
15	14	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)) = ∅)
16		f1f 6720	. . . . . . . . . . . 12 ⊢ (𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)) → 𝑎:((1...𝐴) ∖ (1...𝑁))⟶(𝑆 ∖ (1...𝑁)))
17	16	frnd 6660	. . . . . . . . . . 11 ⊢ (𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)) → ran 𝑎 ⊆ (𝑆 ∖ (1...𝑁)))
18	17	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)))
19	18	ssrind 4195	. . . . . . . . 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)))
20		disjdifr 4424	. . . . . . . . 9 ⊢ ((𝑆 ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅
21	19, 20	sseqtrdi 3976	. . . . . . . 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)) ⊆ ∅)
22		ss0 4353	. . . . . . . 8 ⊢ ((ran 𝑎 ∩ (1...𝑁)) ⊆ ∅ → (ran 𝑎 ∩ (1...𝑁)) = ∅)
23	21, 22	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)) = ∅)
24		f1oun 6783	. . . . . . 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)))
25	11, 13, 15, 23, 24	syl22anc 838	. . . . . 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)))
26		f1of1 6763	. . . . . 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)))
27	25, 26	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)))
28		uncom 4109	. . . . . . 7 ⊢ (((1...𝐴) ∖ (1...𝑁)) ∪ (1...𝑁)) = ((1...𝑁) ∪ ((1...𝐴) ∖ (1...𝑁)))
29		simplrr 777	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → 𝐴 ∈ (ℤ≥‘𝑁))
30		fzss2 13467	. . . . . . . . 9 ⊢ (𝐴 ∈ (ℤ≥‘𝑁) → (1...𝑁) ⊆ (1...𝐴))
31	29, 30	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))
32		undif 4433	. . . . . . . 8 ⊢ ((1...𝑁) ⊆ (1...𝐴) ↔ ((1...𝑁) ∪ ((1...𝐴) ∖ (1...𝑁))) = (1...𝐴))
33	31, 32	sylib 218	. . . . . . 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))
34	28, 33	eqtrid 2776	. . . . . 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))
35		f1eq2 6716	. . . . . 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))))
36	34, 35	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))))
37	27, 36	mpbid 232	. . . 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)))
38	17	difss2d 4090	. . . . . 6 ⊢ (𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)) → ran 𝑎 ⊆ 𝑆)
39	38	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)
40		simplrl 776	. . . . 5 ⊢ ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (1...𝑁) ⊆ 𝑆)
41	39, 40	unssd 4143	. . . 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)
42		f1ss 6725	. . . 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)
43	37, 41, 42	syl2anc 584	. . 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)
44		resundir 5945	. . . 4 ⊢ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ((𝑎 ↾ (1...𝑁)) ∪ (( I ↾ (1...𝑁)) ↾ (1...𝑁)))
45		dmres 5963	. . . . . . . 8 ⊢ dom (𝑎 ↾ (1...𝑁)) = ((1...𝑁) ∩ dom 𝑎)
46		incom 4160	. . . . . . . . 9 ⊢ ((1...𝑁) ∩ dom 𝑎) = (dom 𝑎 ∩ (1...𝑁))
47		f1dm 6724	. . . . . . . . . . . 12 ⊢ (𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)) → dom 𝑎 = ((1...𝐴) ∖ (1...𝑁)))
48	47	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)))
49	48	ineq1d 4170	. . . . . . . . . 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)))
50	49, 14	eqtrdi 2780	. . . . . . . . 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)) = ∅)
51	46, 50	eqtrid 2776	. . . . . . . 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) = ∅)
52	45, 51	eqtrid 2776	. . . . . . 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)) = ∅)
53		relres 5956	. . . . . . . 8 ⊢ Rel (𝑎 ↾ (1...𝑁))
54		reldm0 5870	. . . . . . . 8 ⊢ (Rel (𝑎 ↾ (1...𝑁)) → ((𝑎 ↾ (1...𝑁)) = ∅ ↔ dom (𝑎 ↾ (1...𝑁)) = ∅))
55	53, 54	ax-mp 5	. . . . . . 7 ⊢ ((𝑎 ↾ (1...𝑁)) = ∅ ↔ dom (𝑎 ↾ (1...𝑁)) = ∅)
56	52, 55	sylibr 234	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ≥‘𝑁))) ∧ 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁))) → (𝑎 ↾ (1...𝑁)) = ∅)
57		residm 5961	. . . . . . 7 ⊢ (( I ↾ (1...𝑁)) ↾ (1...𝑁)) = ( I ↾ (1...𝑁))
58	57	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)))
59	56, 58	uneq12d 4120	. . . . 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))))
60		uncom 4109	. . . . . 6 ⊢ (∅ ∪ ( I ↾ (1...𝑁))) = (( I ↾ (1...𝑁)) ∪ ∅)
61		un0 4345	. . . . . 6 ⊢ (( I ↾ (1...𝑁)) ∪ ∅) = ( I ↾ (1...𝑁))
62	60, 61	eqtri 2752	. . . . 5 ⊢ (∅ ∪ ( I ↾ (1...𝑁))) = ( I ↾ (1...𝑁))
63	59, 62	eqtrdi 2780	. . . 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)))
64	44, 63	eqtrid 2776	. . 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)))
65		vex 3440	. . . . 5 ⊢ 𝑎 ∈ V
66		ovex 7382	. . . . . 6 ⊢ (1...𝑁) ∈ V
67		resiexg 7845	. . . . . 6 ⊢ ((1...𝑁) ∈ V → ( I ↾ (1...𝑁)) ∈ V)
68	66, 67	ax-mp 5	. . . . 5 ⊢ ( I ↾ (1...𝑁)) ∈ V
69	65, 68	unex 7680	. . . 4 ⊢ (𝑎 ∪ ( I ↾ (1...𝑁))) ∈ V
70		f1eq1 6715	. . . . 5 ⊢ (𝑐 = (𝑎 ∪ ( I ↾ (1...𝑁))) → (𝑐:(1...𝐴)–1-1→𝑆 ↔ (𝑎 ∪ ( I ↾ (1...𝑁))):(1...𝐴)–1-1→𝑆))
71		reseq1 5924	. . . . . 6 ⊢ (𝑐 = (𝑎 ∪ ( I ↾ (1...𝑁))) → (𝑐 ↾ (1...𝑁)) = ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)))
72	71	eqeq1d 2731	. . . . 5 ⊢ (𝑐 = (𝑎 ∪ ( I ↾ (1...𝑁))) → ((𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)) ↔ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))
73	70, 72	anbi12d 632	. . . 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)))))
74	69, 73	spcev 3561	. . 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))))
75	43, 64, 74	syl2anc 584	. 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))))
76	9, 75	exlimddv 1935	1 ⊢ (((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ≥‘𝑁))) → ∃𝑐(𝑐:(1...𝐴)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3436   ∖ cdif 3900   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284   I cid 5513  dom cdm 5619  ran crn 5620   ↾ cres 5621  Rel wrel 5624  –1-1→wf1 6479  –1-1-onto→wf1o 6481  ‘cfv 6482  (class class class)co 7349  Fincfn 8872  1c1 11010  ℕ₀cn0 12384  ℤ_≥cuz 12735  ...cfz 13410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-n0 12385  df-z 12472  df-uz 12736  df-fz 13411

This theorem is referenced by:  eldioph2b  42756