fin1a2lem6 - Metamath Proof Explorer

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Theorem fin1a2lem6 10402

Description: Lemma for fin1a2 10412. Establish that ω can be broken into two equipollent pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)

**Hypotheses**
Ref	Expression
fin1a2lem.b	⊢ 𝐸 = (𝑥 ∈ ω ↦ (2_o ·_o 𝑥))
fin1a2lem.aa	⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)

**Assertion**
Ref	Expression
fin1a2lem6	⊢ (𝑆 ↾ ran 𝐸):ran 𝐸–1-1-onto→(ω ∖ ran 𝐸)

Proof of Theorem fin1a2lem6 Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fin1a2lem.aa	. . . 4 ⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
2	1	fin1a2lem2 10398	. . 3 ⊢ 𝑆:On–1-1→On
3		fin1a2lem.b	. . . . 5 ⊢ 𝐸 = (𝑥 ∈ ω ↦ (2_o ·_o 𝑥))
4	3	fin1a2lem4 10400	. . . 4 ⊢ 𝐸:ω–1-1→ω
5		f1f 6787	. . . 4 ⊢ (𝐸:ω–1-1→ω → 𝐸:ω⟶ω)
6		frn 6724	. . . . 5 ⊢ (𝐸:ω⟶ω → ran 𝐸 ⊆ ω)
7		omsson 7861	. . . . 5 ⊢ ω ⊆ On
8	6, 7	sstrdi 3994	. . . 4 ⊢ (𝐸:ω⟶ω → ran 𝐸 ⊆ On)
9	4, 5, 8	mp2b 10	. . 3 ⊢ ran 𝐸 ⊆ On
10		f1ores 6847	. . 3 ⊢ ((𝑆:On–1-1→On ∧ ran 𝐸 ⊆ On) → (𝑆 ↾ ran 𝐸):ran 𝐸–1-1-onto→(𝑆 “ ran 𝐸))
11	2, 9, 10	mp2an 690	. 2 ⊢ (𝑆 ↾ ran 𝐸):ran 𝐸–1-1-onto→(𝑆 “ ran 𝐸)
12	9	sseli 3978	. . . . . . . . 9 ⊢ (𝑏 ∈ ran 𝐸 → 𝑏 ∈ On)
13	1	fin1a2lem1 10397	. . . . . . . . 9 ⊢ (𝑏 ∈ On → (𝑆‘𝑏) = suc 𝑏)
14	12, 13	syl 17	. . . . . . . 8 ⊢ (𝑏 ∈ ran 𝐸 → (𝑆‘𝑏) = suc 𝑏)
15	14	eqeq1d 2734	. . . . . . 7 ⊢ (𝑏 ∈ ran 𝐸 → ((𝑆‘𝑏) = 𝑎 ↔ suc 𝑏 = 𝑎))
16	15	rexbiia 3092	. . . . . 6 ⊢ (∃𝑏 ∈ ran 𝐸(𝑆‘𝑏) = 𝑎 ↔ ∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎)
17	4, 5, 6	mp2b 10	. . . . . . . . . . . 12 ⊢ ran 𝐸 ⊆ ω
18	17	sseli 3978	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ran 𝐸 → 𝑏 ∈ ω)
19		peano2 7883	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ω → suc 𝑏 ∈ ω)
20	18, 19	syl 17	. . . . . . . . . 10 ⊢ (𝑏 ∈ ran 𝐸 → suc 𝑏 ∈ ω)
21	3	fin1a2lem5 10401	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ω → (𝑏 ∈ ran 𝐸 ↔ ¬ suc 𝑏 ∈ ran 𝐸))
22	21	biimpd 228	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ω → (𝑏 ∈ ran 𝐸 → ¬ suc 𝑏 ∈ ran 𝐸))
23	18, 22	mpcom 38	. . . . . . . . . 10 ⊢ (𝑏 ∈ ran 𝐸 → ¬ suc 𝑏 ∈ ran 𝐸)
24	20, 23	jca 512	. . . . . . . . 9 ⊢ (𝑏 ∈ ran 𝐸 → (suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸))
25		eleq1 2821	. . . . . . . . . 10 ⊢ (suc 𝑏 = 𝑎 → (suc 𝑏 ∈ ω ↔ 𝑎 ∈ ω))
26		eleq1 2821	. . . . . . . . . . 11 ⊢ (suc 𝑏 = 𝑎 → (suc 𝑏 ∈ ran 𝐸 ↔ 𝑎 ∈ ran 𝐸))
27	26	notbid 317	. . . . . . . . . 10 ⊢ (suc 𝑏 = 𝑎 → (¬ suc 𝑏 ∈ ran 𝐸 ↔ ¬ 𝑎 ∈ ran 𝐸))
28	25, 27	anbi12d 631	. . . . . . . . 9 ⊢ (suc 𝑏 = 𝑎 → ((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸)))
29	24, 28	syl5ibcom 244	. . . . . . . 8 ⊢ (𝑏 ∈ ran 𝐸 → (suc 𝑏 = 𝑎 → (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸)))
30	29	rexlimiv 3148	. . . . . . 7 ⊢ (∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎 → (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))
31		peano1 7881	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ ω
32	3	fin1a2lem3 10399	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ ω → (𝐸‘∅) = (2_o ·_o ∅))
33	31, 32	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝐸‘∅) = (2_o ·_o ∅)
34		2on 8482	. . . . . . . . . . . . . 14 ⊢ 2_o ∈ On
35		om0 8519	. . . . . . . . . . . . . 14 ⊢ (2_o ∈ On → (2_o ·_o ∅) = ∅)
36	34, 35	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (2_o ·_o ∅) = ∅
37	33, 36	eqtri 2760	. . . . . . . . . . . 12 ⊢ (𝐸‘∅) = ∅
38		f1fun 6789	. . . . . . . . . . . . . 14 ⊢ (𝐸:ω–1-1→ω → Fun 𝐸)
39	4, 38	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun 𝐸
40		f1dm 6791	. . . . . . . . . . . . . . 15 ⊢ (𝐸:ω–1-1→ω → dom 𝐸 = ω)
41	4, 40	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ dom 𝐸 = ω
42	31, 41	eleqtrri 2832	. . . . . . . . . . . . 13 ⊢ ∅ ∈ dom 𝐸
43		fvelrn 7078	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐸 ∧ ∅ ∈ dom 𝐸) → (𝐸‘∅) ∈ ran 𝐸)
44	39, 42, 43	mp2an 690	. . . . . . . . . . . 12 ⊢ (𝐸‘∅) ∈ ran 𝐸
45	37, 44	eqeltrri 2830	. . . . . . . . . . 11 ⊢ ∅ ∈ ran 𝐸
46		eleq1 2821	. . . . . . . . . . 11 ⊢ (𝑎 = ∅ → (𝑎 ∈ ran 𝐸 ↔ ∅ ∈ ran 𝐸))
47	45, 46	mpbiri 257	. . . . . . . . . 10 ⊢ (𝑎 = ∅ → 𝑎 ∈ ran 𝐸)
48	47	necon3bi 2967	. . . . . . . . 9 ⊢ (¬ 𝑎 ∈ ran 𝐸 → 𝑎 ≠ ∅)
49		nnsuc 7875	. . . . . . . . 9 ⊢ ((𝑎 ∈ ω ∧ 𝑎 ≠ ∅) → ∃𝑏 ∈ ω 𝑎 = suc 𝑏)
50	48, 49	sylan2 593	. . . . . . . 8 ⊢ ((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) → ∃𝑏 ∈ ω 𝑎 = suc 𝑏)
51		eleq1 2821	. . . . . . . . . . . . 13 ⊢ (𝑎 = suc 𝑏 → (𝑎 ∈ ω ↔ suc 𝑏 ∈ ω))
52		eleq1 2821	. . . . . . . . . . . . . 14 ⊢ (𝑎 = suc 𝑏 → (𝑎 ∈ ran 𝐸 ↔ suc 𝑏 ∈ ran 𝐸))
53	52	notbid 317	. . . . . . . . . . . . 13 ⊢ (𝑎 = suc 𝑏 → (¬ 𝑎 ∈ ran 𝐸 ↔ ¬ suc 𝑏 ∈ ran 𝐸))
54	51, 53	anbi12d 631	. . . . . . . . . . . 12 ⊢ (𝑎 = suc 𝑏 → ((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ↔ (suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸)))
55	54	anbi1d 630	. . . . . . . . . . 11 ⊢ (𝑎 = suc 𝑏 → (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) ↔ ((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ ω)))
56		simplr 767	. . . . . . . . . . . 12 ⊢ (((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → ¬ suc 𝑏 ∈ ran 𝐸)
57	21	adantl 482	. . . . . . . . . . . 12 ⊢ (((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → (𝑏 ∈ ran 𝐸 ↔ ¬ suc 𝑏 ∈ ran 𝐸))
58	56, 57	mpbird 256	. . . . . . . . . . 11 ⊢ (((suc 𝑏 ∈ ω ∧ ¬ suc 𝑏 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → 𝑏 ∈ ran 𝐸)
59	55, 58	syl6bi 252	. . . . . . . . . 10 ⊢ (𝑎 = suc 𝑏 → (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → 𝑏 ∈ ran 𝐸))
60	59	com12 32	. . . . . . . . 9 ⊢ (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ 𝑏 ∈ ω) → (𝑎 = suc 𝑏 → 𝑏 ∈ ran 𝐸))
61	60	impr 455	. . . . . . . 8 ⊢ (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → 𝑏 ∈ ran 𝐸)
62		simprr 771	. . . . . . . . 9 ⊢ (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → 𝑎 = suc 𝑏)
63	62	eqcomd 2738	. . . . . . . 8 ⊢ (((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) ∧ (𝑏 ∈ ω ∧ 𝑎 = suc 𝑏)) → suc 𝑏 = 𝑎)
64	50, 61, 63	reximssdv 3172	. . . . . . 7 ⊢ ((𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸) → ∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎)
65	30, 64	impbii 208	. . . . . 6 ⊢ (∃𝑏 ∈ ran 𝐸 suc 𝑏 = 𝑎 ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))
66	16, 65	bitri 274	. . . . 5 ⊢ (∃𝑏 ∈ ran 𝐸(𝑆‘𝑏) = 𝑎 ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))
67		f1fn 6788	. . . . . . 7 ⊢ (𝑆:On–1-1→On → 𝑆 Fn On)
68	2, 67	ax-mp 5	. . . . . 6 ⊢ 𝑆 Fn On
69		fvelimab 6964	. . . . . 6 ⊢ ((𝑆 Fn On ∧ ran 𝐸 ⊆ On) → (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ ∃𝑏 ∈ ran 𝐸(𝑆‘𝑏) = 𝑎))
70	68, 9, 69	mp2an 690	. . . . 5 ⊢ (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ ∃𝑏 ∈ ran 𝐸(𝑆‘𝑏) = 𝑎)
71		eldif 3958	. . . . 5 ⊢ (𝑎 ∈ (ω ∖ ran 𝐸) ↔ (𝑎 ∈ ω ∧ ¬ 𝑎 ∈ ran 𝐸))
72	66, 70, 71	3bitr4i 302	. . . 4 ⊢ (𝑎 ∈ (𝑆 “ ran 𝐸) ↔ 𝑎 ∈ (ω ∖ ran 𝐸))
73	72	eqriv 2729	. . 3 ⊢ (𝑆 “ ran 𝐸) = (ω ∖ ran 𝐸)
74		f1oeq3 6823	. . 3 ⊢ ((𝑆 “ ran 𝐸) = (ω ∖ ran 𝐸) → ((𝑆 ↾ ran 𝐸):ran 𝐸–1-1-onto→(𝑆 “ ran 𝐸) ↔ (𝑆 ↾ ran 𝐸):ran 𝐸–1-1-onto→(ω ∖ ran 𝐸)))
75	73, 74	ax-mp 5	. 2 ⊢ ((𝑆 ↾ ran 𝐸):ran 𝐸–1-1-onto→(𝑆 “ ran 𝐸) ↔ (𝑆 ↾ ran 𝐸):ran 𝐸–1-1-onto→(ω ∖ ran 𝐸))
76	11, 75	mpbi 229	1 ⊢ (𝑆 ↾ ran 𝐸):ran 𝐸–1-1-onto→(ω ∖ ran 𝐸)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∃wrex 3070 ∖ cdif 3945 ⊆ wss 3948 ∅c0 4322 ↦ cmpt 5231 dom cdm 5676 ran crn 5677 ↾ cres 5678 “ cima 5679 Oncon0 6364 suc csuc 6366 Fun wfun 6537 Fn wfn 6538 ⟶wf 6539 –1-1→wf1 6540 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7411 ωcom 7857 2_oc2o 8462 ·_o comu 8466

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7727

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-omul 8473

This theorem is referenced by: fin1a2lem7 10403

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