Theorem infxpenc2 10047

Description: Existence form of infxpenc 10043. A "uniform" or "canonical" version of infxpen 10039, 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 9731	. 2 ⊢ (𝐴 ∈ On → ∃𝑛∀𝑥 ∈ 𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦)))
2		df-2o 8488	. . . . . . . 8 ⊢ 2o = suc 1o
3	2	oveq2i 7430	. . . . . . 7 ⊢ (ω ↑o 2o) = (ω ↑o suc 1o)
4		omelon 9671	. . . . . . . 8 ⊢ ω ∈ On
5		1on 8499	. . . . . . . 8 ⊢ 1o ∈ On
6		oesuc 8548	. . . . . . . 8 ⊢ ((ω ∈ On ∧ 1o ∈ On) → (ω ↑o suc 1o) = ((ω ↑o 1o) ·o ω))
7	4, 5, 6	mp2an 690	. . . . . . 7 ⊢ (ω ↑o suc 1o) = ((ω ↑o 1o) ·o ω)
8		oe1 8565	. . . . . . . . 9 ⊢ (ω ∈ On → (ω ↑o 1o) = ω)
9	4, 8	ax-mp 5	. . . . . . . 8 ⊢ (ω ↑o 1o) = ω
10	9	oveq1i 7429	. . . . . . 7 ⊢ ((ω ↑o 1o) ·o ω) = (ω ·o ω)
11	3, 7, 10	3eqtri 2757	. . . . . 6 ⊢ (ω ↑o 2o) = (ω ·o ω)
12		omxpen 9099	. . . . . . 7 ⊢ ((ω ∈ On ∧ ω ∈ On) → (ω ·o ω) ≈ (ω × ω))
13	4, 4, 12	mp2an 690	. . . . . 6 ⊢ (ω ·o ω) ≈ (ω × ω)
14	11, 13	eqbrtri 5170	. . . . 5 ⊢ (ω ↑o 2o) ≈ (ω × ω)
15		xpomen 10040	. . . . 5 ⊢ (ω × ω) ≈ ω
16	14, 15	entri 9029	. . . 4 ⊢ (ω ↑o 2o) ≈ ω
17	16	a1i 11	. . 3 ⊢ (𝐴 ∈ On → (ω ↑o 2o) ≈ ω)
18		bren 8974	. . 3 ⊢ ((ω ↑o 2o) ≈ ω ↔ ∃𝑓 𝑓:(ω ↑o 2o)–1-1-onto→ω)
19	17, 18	sylib 217	. 2 ⊢ (𝐴 ∈ On → ∃𝑓 𝑓:(ω ↑o 2o)–1-1-onto→ω)
20		exdistrv 1951	. . 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 481	. . . . . 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 769	. . . . . . 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 4003	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → (ω ⊆ 𝑥 ↔ ω ⊆ 𝑏))
24		oveq2 7427	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (ω ↑o 𝑦) = (ω ↑o 𝑤))
25	24	f1oeq3d 6835	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → ((𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑦) ↔ (𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑤)))
26	25	cbvrexvw 3225	. . . . . . . . . 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 6896	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑏 → (𝑛‘𝑥) = (𝑛‘𝑏))
28	27	f1oeq1d 6833	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑏 → ((𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑤) ↔ (𝑛‘𝑏):𝑥–1-1-onto→(ω ↑o 𝑤)))
29		f1oeq2 6827	. . . . . . . . . . . 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 278	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑏 → ((𝑛‘𝑥):𝑥–1-1-onto→(ω ↑o 𝑤) ↔ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
31	30	rexbidv 3168	. . . . . . . . . 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 282	. . . . . . . . 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 343	. . . . . . . 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 3224	. . . . . . 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 217	. . . . . 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 7427	. . . . . . . . 9 ⊢ (𝑏 = 𝑧 → (ω ↑o 𝑏) = (ω ↑o 𝑧))
37	36	cbvmptv 5262	. . . . . . . 8 ⊢ (𝑏 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑏)) = (𝑧 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑧))
38	37	cnveqi 5877	. . . . . . 7 ⊢ ◡(𝑏 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑏)) = ◡(𝑧 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑧))
39	38	fveq1i 6897	. . . . . 6 ⊢ (◡(𝑏 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑏))‘ran (𝑛‘𝑏)) = (◡(𝑧 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑧))‘ran (𝑛‘𝑏))
40		2on 8501	. . . . . . . . . 10 ⊢ 2o ∈ On
41		peano1 7895	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
42		oen0 8607	. . . . . . . . . . 11 ⊢ (((ω ∈ On ∧ 2o ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 2o))
43	41, 42	mpan2 689	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ 2o ∈ On) → ∅ ∈ (ω ↑o 2o))
44	4, 40, 43	mp2an 690	. . . . . . . . 9 ⊢ ∅ ∈ (ω ↑o 2o)
45		eqid 2725	. . . . . . . . . 10 ⊢ (𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩})) = (𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (◡𝑓‘∅)})) ∪ {⟨∅, (◡𝑓‘∅)⟩, ⟨(◡𝑓‘∅), ∅⟩}))
46	45	fveqf1o 7311	. . . . . . . . 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 1449	. . . . . . . 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 727	. . . . . . 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 493	. . . . . 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 494	. . . . . 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 10046	. . . . 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 411	. . . 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 1929	. . 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 242	. 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 697	1 ⊢ (𝐴 ∈ On → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∀wral 3050  ∃wrex 3059   ∖ cdif 3941   ∪ cun 3942   ⊆ wss 3944  ∅c0 4322  {cpr 4632  ⟨cop 4636   class class class wbr 5149   ↦ cmpt 5232   I cid 5575   × cxp 5676  ^◡ccnv 5677  ran crn 5679   ↾ cres 5680   ∘ ccom 5682  Oncon0 6371  suc csuc 6373  –1-1-onto→wf1o 6548  ‘cfv 6549  (class class class)co 7419  ωcom 7871  1_oc1o 8480  2_oc2o 8481   ·_o comu 8485   ↑_o coe 8486   ≈ cen 8961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-supp 8166  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-seqom 8469  df-1o 8487  df-2o 8488  df-oadd 8491  df-omul 8492  df-oexp 8493  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9388  df-oi 9535  df-cnf 9687  df-card 9964

This theorem is referenced by:  pwfseq  10689