Theorem eldioph2lem2 43347

Description: Lemma for eldioph2 43348. 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 778	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ≥‘𝑁))) → ¬ 𝑆 ∈ Fin)
2		fzfi 13987	. . . 4 ⊢ (1...𝑁) ∈ Fin
3		difinf 9257	. . . 4 ⊢ ((¬ 𝑆 ∈ Fin ∧ (1...𝑁) ∈ Fin) → ¬ (𝑆 ∖ (1...𝑁)) ∈ Fin)
4	1, 2, 3	sylancl 595	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ≥‘𝑁))) → ¬ (𝑆 ∖ (1...𝑁)) ∈ Fin)
5		fzfi 13987	. . . 4 ⊢ (1...𝐴) ∈ Fin
6		diffi 9145	. . . 4 ⊢ ((1...𝐴) ∈ Fin → ((1...𝐴) ∖ (1...𝑁)) ∈ Fin)
7	5, 6	ax-mp 5	. . 3 ⊢ ((1...𝐴) ∖ (1...𝑁)) ∈ Fin
8		isinffi 9952	. . 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 595	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ ¬ 𝑆 ∈ Fin) ∧ ((1...𝑁) ⊆ 𝑆 ∧ 𝐴 ∈ (ℤ≥‘𝑁))) → ∃𝑎 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)))
10		f1f1orn 6820	. . . . . . . 8 ⊢ (𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)) → 𝑎:((1...𝐴) ∖ (1...𝑁))–1-1-onto→ran 𝑎)
11	10	adantl 485	. . . . . . 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 6847	. . . . . . . 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 4429	. . . . . . . 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 6762	. . . . . . . . . . . 12 ⊢ (𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)) → 𝑎:((1...𝐴) ∖ (1...𝑁))⟶(𝑆 ∖ (1...𝑁)))
17	16	frnd 6702	. . . . . . . . . . 11 ⊢ (𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)) → ran 𝑎 ⊆ (𝑆 ∖ (1...𝑁)))
18	17	adantl 485	. . . . . . . . . 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 4197	. . . . . . . . 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 4429	. . . . . . . . 9 ⊢ ((𝑆 ∖ (1...𝑁)) ∩ (1...𝑁)) = ∅
21	19, 20	sseqtrdi 3978	. . . . . . . 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 4358	. . . . . . . 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 6828	. . . . . . 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 849	. . . . . 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 6807	. . . . . 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 4113	. . . . . . 7 ⊢ (((1...𝐴) ∖ (1...𝑁)) ∪ (1...𝑁)) = ((1...𝑁) ∪ ((1...𝐴) ∖ (1...𝑁)))
29		simplrr 787	. . . . . . . . 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 13571	. . . . . . . . 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 4438	. . . . . . . 8 ⊢ ((1...𝑁) ⊆ (1...𝐴) ↔ ((1...𝑁) ∪ ((1...𝐴) ∖ (1...𝑁))) = (1...𝐴))
33	31, 32	sylib 220	. . . . . . 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 2811	. . . . . 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 6758	. . . . . 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 234	. . . 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 4094	. . . . . 6 ⊢ (𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)) → ran 𝑎 ⊆ 𝑆)
39	38	adantl 485	. . . . 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 786	. . . . 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 4146	. . . 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 6769	. . . 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 593	. . 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 5982	. . . 4 ⊢ ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)) = ((𝑎 ↾ (1...𝑁)) ∪ (( I ↾ (1...𝑁)) ↾ (1...𝑁)))
45		dmres 6000	. . . . . . . 8 ⊢ dom (𝑎 ↾ (1...𝑁)) = ((1...𝑁) ∩ dom 𝑎)
46		incom 4163	. . . . . . . . 9 ⊢ ((1...𝑁) ∩ dom 𝑎) = (dom 𝑎 ∩ (1...𝑁))
47		f1dm 6768	. . . . . . . . . . . 12 ⊢ (𝑎:((1...𝐴) ∖ (1...𝑁))–1-1→(𝑆 ∖ (1...𝑁)) → dom 𝑎 = ((1...𝐴) ∖ (1...𝑁)))
48	47	adantl 485	. . . . . . . . . . 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 4173	. . . . . . . . . 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 2815	. . . . . . . . 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 2811	. . . . . . . 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 2811	. . . . . . 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 5993	. . . . . . . 8 ⊢ Rel (𝑎 ↾ (1...𝑁))
54		reldm0 5906	. . . . . . . 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 236	. . . . . 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 5998	. . . . . . 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 4124	. . . . 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 4113	. . . . . 6 ⊢ (∅ ∪ ( I ↾ (1...𝑁))) = (( I ↾ (1...𝑁)) ∪ ∅)
61		un0 4350	. . . . . 6 ⊢ (( I ↾ (1...𝑁)) ∪ ∅) = ( I ↾ (1...𝑁))
62	60, 61	eqtri 2787	. . . . 5 ⊢ (∅ ∪ ( I ↾ (1...𝑁))) = ( I ↾ (1...𝑁))
63	59, 62	eqtrdi 2815	. . . 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 2811	. . 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 3460	. . . . 5 ⊢ 𝑎 ∈ V
66		ovex 7431	. . . . . 6 ⊢ (1...𝑁) ∈ V
67		resiexg 7895	. . . . . 6 ⊢ ((1...𝑁) ∈ V → ( I ↾ (1...𝑁)) ∈ V)
68	66, 67	ax-mp 5	. . . . 5 ⊢ ( I ↾ (1...𝑁)) ∈ V
69	65, 68	unex 7729	. . . 4 ⊢ (𝑎 ∪ ( I ↾ (1...𝑁))) ∈ V
70		f1eq1 6757	. . . . 5 ⊢ (𝑐 = (𝑎 ∪ ( I ↾ (1...𝑁))) → (𝑐:(1...𝐴)–1-1→𝑆 ↔ (𝑎 ∪ ( I ↾ (1...𝑁))):(1...𝐴)–1-1→𝑆))
71		reseq1 5961	. . . . . 6 ⊢ (𝑐 = (𝑎 ∪ ( I ↾ (1...𝑁))) → (𝑐 ↾ (1...𝑁)) = ((𝑎 ∪ ( I ↾ (1...𝑁))) ↾ (1...𝑁)))
72	71	eqeq1d 2766	. . . . 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 641	. . . 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 3567	. . 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 593	. 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 1957	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 208   ∧ wa 399   = wceq 1562  ∃wex 1801   ∈ wcel 2144  Vcvv 3456   ∖ cdif 3903   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287   I cid 5543  dom cdm 5649  ran crn 5650   ↾ cres 5651  Rel wrel 5654  –1-1→wf1 6520  –1-1-onto→wf1o 6522  ‘cfv 6523  (class class class)co 7398  Fincfn 8929  1c1 11076  ℕ₀cn0 12483  ℤ_≥cuz 12841  ...cfz 13514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-er 8680  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-card 9899  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-nn 12213  df-n0 12484  df-z 12571  df-uz 12842  df-fz 13515

This theorem is referenced by:  eldioph2b  43349