Theorem infxpenc2 10002

Description: Existence form of infxpenc 9998. A "uniform" or "canonical" version of infxpen 9994, asserting the existence of a single function 𝑔 that simultaneously demonstrates product idempotence of all ordinals below a given bound. (Contributed by Mario Carneiro, 30-May-2015.)

Assertion

Ref	Expression
infxpenc2	⊢ (𝐴 ∈ On → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))

Distinct variable group:   𝑔,𝑏,𝐴

Proof of Theorem infxpenc2

Dummy variables 𝑓 𝑛 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnfcom3c 9671	. 2 ⊢ (𝐴 ∈ On → ∃𝑛∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)))
2		df-2o 8450	. . . . . . . 8 ⊢ 2o = suc 1o
3	2	oveq2i 7421	. . . . . . 7 ⊢ (ω ↑o 2o) = (ω ↑o suc 1o)
4		omelon 9611	. . . . . . . 8 ⊢ ω ∈ On
5		1on 8462	. . . . . . . 8 ⊢ 1o ∈ On
6		oesuc 8508	. . . . . . . 8 ⊢ ((ω ∈ On ∧ 1o ∈ On) → (ω ↑o suc 1o) = ((ω ↑o 1o) ·o ω))
7	4, 5, 6	mp2an 704	. . . . . . 7 ⊢ (ω ↑o suc 1o) = ((ω ↑o 1o) ·o ω)
8		oe1 8525	. . . . . . . . 9 ⊢ (ω ∈ On → (ω ↑o 1o) = ω)
9	4, 8	ax-mp 5	. . . . . . . 8 ⊢ (ω ↑o 1o) = ω
10	9	oveq1i 7420	. . . . . . 7 ⊢ ((ω ↑o 1o) ·o ω) = (ω ·o ω)
11	3, 7, 10	3eqtri 2790	. . . . . 6 ⊢ (ω ↑o 2o) = (ω ·o ω)
12		omxpen 9063	. . . . . . 7 ⊢ ((ω ∈ On ∧ ω ∈ On) → (ω ·o ω) ≈ (ω × ω))
13	4, 4, 12	mp2an 704	. . . . . 6 ⊢ (ω ·o ω) ≈ (ω × ω)
14	11, 13	eqbrtri 5132	. . . . 5 ⊢ (ω ↑o 2o) ≈ (ω × ω)
15		xpomen 9995	. . . . 5 ⊢ (ω × ω) ≈ ω
16	14, 15	entri 9001	. . . 4 ⊢ (ω ↑o 2o) ≈ ω
17	16	a1i 11	. . 3 ⊢ (𝐴 ∈ On → (ω ↑o 2o) ≈ ω)
18		bren 8949	. . 3 ⊢ ((ω ↑o 2o) ≈ ω ↔ ∃𝑓 𝑓:(ω ↑o 2o)–1-1-onto→ω)
19	17, 18	sylib 221	. 2 ⊢ (𝐴 ∈ On → ∃𝑓 𝑓:(ω ↑o 2o)–1-1-onto→ω)
20		exdistrv 1985	. . 3 ⊢ (∃𝑛∃𝑓(∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ 𝑓:(ω ↑o 2o)–1-1-onto→ω) ↔ (∃𝑛∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ ∃𝑓 𝑓:(ω ↑o 2o)–1-1-onto→ω))
21		simpl 487	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ 𝑓:(ω ↑o 2o)–1-1-onto→ω)) → 𝐴 ∈ On)
22		simprl 782	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ 𝑓:(ω ↑o 2o)–1-1-onto→ω)) → ∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)))
23		sseq2 3963	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → (ω ⊆ 𝑥 ↔ ω ⊆ 𝑏))
24		oveq2 7418	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (ω ↑o 𝑦) = (ω ↑o 𝑤))
25	24	f1oeq3d 6817	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → ((𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦) ↔ (𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑤)))
26	25	cbvrexvw 3244	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦) ↔ ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑤))
27		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑏 → (𝑛‘𝑥) = (𝑛‘𝑏))
28	27	f1oeq1d 6815	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑏 → ((𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑤) ↔ (𝑛‘𝑏):𝑥–1-1-onto→(ω ↑o 𝑤)))
29		f1oeq2 6809	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑏 → ((𝑛‘𝑏):𝑥–1-1-onto→(ω ↑o 𝑤) ↔ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
30	28, 29	bitrd 282	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑏 → ((𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑤) ↔ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
31	30	rexbidv 3189	. . . . . . . . . 10 ⊢ (𝑥 = 𝑏 → (∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑤) ↔ ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
32	26, 31	bitrid 286	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → (∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦) ↔ ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
33	23, 32	imbi12d 347	. . . . . . . 8 ⊢ (𝑥 = 𝑏 → ((ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ↔ (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))))
34	33	cbvralvw 3243	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ↔ ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
35	22, 34	sylib 221	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ 𝑓:(ω ↑o 2o)–1-1-onto→ω)) → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
36		oveq2 7418	. . . . . . . . 9 ⊢ (𝑏 = 𝑧 → (ω ↑o 𝑏) = (ω ↑o 𝑧))
37	36	cbvmptv 5215	. . . . . . . 8 ⊢ (𝑏 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑏)) = (𝑧 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑧))
38	37	cnveqi 5860	. . . . . . 7 ⊢ ◡(𝑏 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑏)) = ◡(𝑧 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑧))
39	38	fveq1i 6882	. . . . . 6 ⊢ (◡(𝑏 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑏))‘ran (𝑛‘𝑏)) = (◡(𝑧 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑧))‘ran (𝑛‘𝑏))
40		2on 8463	. . . . . . . . . 10 ⊢ 2o ∈ On
41		peano1 7881	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
42		oen0 8568	. . . . . . . . . . 11 ⊢ (((ω ∈ On ∧ 2o ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 2o))
43	41, 42	mpan2 703	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ 2o ∈ On) → ∅ ∈ (ω ↑o 2o))
44	4, 40, 43	mp2an 704	. . . . . . . . 9 ⊢ ∅ ∈ (ω ↑o 2o)
45		eqid 2763	. . . . . . . . . 10 ⊢ (𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩})) = (𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩}))
46	45	fveqf1o 7300	. . . . . . . . 9 ⊢ ((𝑓:(ω ↑o 2o)–1-1-onto→ω ∧ ∅ ∈ (ω ↑o 2o) ∧ ∅ ∈ ω) → ((𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩})):(ω ↑o 2o)–1-1-onto→ω ∧ ((𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩}))‘∅) = ∅))
47	44, 41, 46	mp3an23 1482	. . . . . . . 8 ⊢ (𝑓:(ω ↑o 2o)–1-1-onto→ω → ((𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩})):(ω ↑o 2o)–1-1-onto→ω ∧ ((𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩}))‘∅) = ∅))
48	47	ad2antll 741	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ 𝑓:(ω ↑o 2o)–1-1-onto→ω)) → ((𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩})):(ω ↑o 2o)–1-1-onto→ω ∧ ((𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩}))‘∅) = ∅))
49	48	simpld 499	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ 𝑓:(ω ↑o 2o)–1-1-onto→ω)) → (𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩})):(ω ↑o 2o)–1-1-onto→ω)
50	48	simprd 500	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ 𝑓:(ω ↑o 2o)–1-1-onto→ω)) → ((𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩}))‘∅) = ∅)
51	21, 35, 39, 49, 50	infxpenc2lem3 10001	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ 𝑓:(ω ↑o 2o)–1-1-onto→ω)) → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))
52	51	ex 417	. . . 4 ⊢ (𝐴 ∈ On → ((∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ 𝑓:(ω ↑o 2o)–1-1-onto→ω) → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)))
53	52	exlimdvv 1964	. . 3 ⊢ (𝐴 ∈ On → (∃𝑛∃𝑓(∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ 𝑓:(ω ↑o 2o)–1-1-onto→ω) → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)))
54	20, 53	biimtrrid 246	. 2 ⊢ (𝐴 ∈ On → ((∃𝑛∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)) ∧ ∃𝑓 𝑓:(ω ↑o 2o)–1-1-onto→ω) → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)))
55	1, 19, 54	mp2and 711	1 ⊢ (𝐴 ∈ On → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ∖ cdif 3902   ∪ cun 3903   ⊆ wss 3905  ∅c0 4286  {cpr 4591  ⟨cop 4595   class class class wbr 5109   ↦ cmpt 5192   I cid 5555   × cxp 5659  ^◡ccnv 5660  ran crn 5662   ↾ cres 5663   ∘ ccom 5665  Oncon0 6360  suc csuc 6362  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410  ωcom 7858  1_oc1o 8442  2_oc2o 8443   ·_o comu 8447   ↑_o coe 8448   ≈ cen 8936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9606

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-supp 8153  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-seqom 8431  df-1o 8449  df-2o 8450  df-oadd 8453  df-omul 8454  df-oexp 8455  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9318  df-oi 9468  df-cnf 9627  df-card 9921

This theorem is referenced by:  pwfseq  10644