Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  infxpenc2 Structured version   Visualization version   GIF version

Theorem infxpenc2 8728
 Description: Existence form of infxpenc 8724. A "uniform" or "canonical" version of infxpen 8720, 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 → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → (𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
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.
StepHypRef Expression
1 cnfcom3c 8486 . 2 (𝐴 ∈ On → ∃𝑛𝑥𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦)))
2 df-2o 7448 . . . . . . . 8 2𝑜 = suc 1𝑜
32oveq2i 6560 . . . . . . 7 (ω ↑𝑜 2𝑜) = (ω ↑𝑜 suc 1𝑜)
4 omelon 8426 . . . . . . . 8 ω ∈ On
5 1on 7454 . . . . . . . 8 1𝑜 ∈ On
6 oesuc 7494 . . . . . . . 8 ((ω ∈ On ∧ 1𝑜 ∈ On) → (ω ↑𝑜 suc 1𝑜) = ((ω ↑𝑜 1𝑜) ·𝑜 ω))
74, 5, 6mp2an 704 . . . . . . 7 (ω ↑𝑜 suc 1𝑜) = ((ω ↑𝑜 1𝑜) ·𝑜 ω)
8 oe1 7511 . . . . . . . . 9 (ω ∈ On → (ω ↑𝑜 1𝑜) = ω)
94, 8ax-mp 5 . . . . . . . 8 (ω ↑𝑜 1𝑜) = ω
109oveq1i 6559 . . . . . . 7 ((ω ↑𝑜 1𝑜) ·𝑜 ω) = (ω ·𝑜 ω)
113, 7, 103eqtri 2636 . . . . . 6 (ω ↑𝑜 2𝑜) = (ω ·𝑜 ω)
12 omxpen 7947 . . . . . . 7 ((ω ∈ On ∧ ω ∈ On) → (ω ·𝑜 ω) ≈ (ω × ω))
134, 4, 12mp2an 704 . . . . . 6 (ω ·𝑜 ω) ≈ (ω × ω)
1411, 13eqbrtri 4604 . . . . 5 (ω ↑𝑜 2𝑜) ≈ (ω × ω)
15 xpomen 8721 . . . . 5 (ω × ω) ≈ ω
1614, 15entri 7896 . . . 4 (ω ↑𝑜 2𝑜) ≈ ω
1716a1i 11 . . 3 (𝐴 ∈ On → (ω ↑𝑜 2𝑜) ≈ ω)
18 bren 7850 . . 3 ((ω ↑𝑜 2𝑜) ≈ ω ↔ ∃𝑓 𝑓:(ω ↑𝑜 2𝑜)–1-1-onto→ω)
1917, 18sylib 207 . 2 (𝐴 ∈ On → ∃𝑓 𝑓:(ω ↑𝑜 2𝑜)–1-1-onto→ω)
20 eeanv 2170 . . 3 (∃𝑛𝑓(∀𝑥𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦)) ∧ 𝑓:(ω ↑𝑜 2𝑜)–1-1-onto→ω) ↔ (∃𝑛𝑥𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦)) ∧ ∃𝑓 𝑓:(ω ↑𝑜 2𝑜)–1-1-onto→ω))
21 simpl 472 . . . . . 6 ((𝐴 ∈ On ∧ (∀𝑥𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦)) ∧ 𝑓:(ω ↑𝑜 2𝑜)–1-1-onto→ω)) → 𝐴 ∈ On)
22 simprl 790 . . . . . . 7 ((𝐴 ∈ On ∧ (∀𝑥𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦)) ∧ 𝑓:(ω ↑𝑜 2𝑜)–1-1-onto→ω)) → ∀𝑥𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦)))
23 sseq2 3590 . . . . . . . . 9 (𝑥 = 𝑏 → (ω ⊆ 𝑥 ↔ ω ⊆ 𝑏))
24 oveq2 6557 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (ω ↑𝑜 𝑦) = (ω ↑𝑜 𝑤))
25 f1oeq3 6042 . . . . . . . . . . . 12 ((ω ↑𝑜 𝑦) = (ω ↑𝑜 𝑤) → ((𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦) ↔ (𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑤)))
2624, 25syl 17 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦) ↔ (𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑤)))
2726cbvrexv 3148 . . . . . . . . . 10 (∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦) ↔ ∃𝑤 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑤))
28 fveq2 6103 . . . . . . . . . . . . 13 (𝑥 = 𝑏 → (𝑛𝑥) = (𝑛𝑏))
29 f1oeq1 6040 . . . . . . . . . . . . 13 ((𝑛𝑥) = (𝑛𝑏) → ((𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑤) ↔ (𝑛𝑏):𝑥1-1-onto→(ω ↑𝑜 𝑤)))
3028, 29syl 17 . . . . . . . . . . . 12 (𝑥 = 𝑏 → ((𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑤) ↔ (𝑛𝑏):𝑥1-1-onto→(ω ↑𝑜 𝑤)))
31 f1oeq2 6041 . . . . . . . . . . . 12 (𝑥 = 𝑏 → ((𝑛𝑏):𝑥1-1-onto→(ω ↑𝑜 𝑤) ↔ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
3230, 31bitrd 267 . . . . . . . . . . 11 (𝑥 = 𝑏 → ((𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑤) ↔ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
3332rexbidv 3034 . . . . . . . . . 10 (𝑥 = 𝑏 → (∃𝑤 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑤) ↔ ∃𝑤 ∈ (On ∖ 1𝑜)(𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
3427, 33syl5bb 271 . . . . . . . . 9 (𝑥 = 𝑏 → (∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦) ↔ ∃𝑤 ∈ (On ∖ 1𝑜)(𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
3523, 34imbi12d 333 . . . . . . . 8 (𝑥 = 𝑏 → ((ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦)) ↔ (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))))
3635cbvralv 3147 . . . . . . 7 (∀𝑥𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦)) ↔ ∀𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
3722, 36sylib 207 . . . . . 6 ((𝐴 ∈ On ∧ (∀𝑥𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦)) ∧ 𝑓:(ω ↑𝑜 2𝑜)–1-1-onto→ω)) → ∀𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
38 oveq2 6557 . . . . . . . . 9 (𝑏 = 𝑧 → (ω ↑𝑜 𝑏) = (ω ↑𝑜 𝑧))
3938cbvmptv 4678 . . . . . . . 8 (𝑏 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑏)) = (𝑧 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑧))
4039cnveqi 5219 . . . . . . 7 (𝑏 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑏)) = (𝑧 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑧))
4140fveq1i 6104 . . . . . 6 ((𝑏 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑏))‘ran (𝑛𝑏)) = ((𝑧 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑧))‘ran (𝑛𝑏))
42 2on 7455 . . . . . . . . . 10 2𝑜 ∈ On
43 peano1 6977 . . . . . . . . . . 11 ∅ ∈ ω
44 oen0 7553 . . . . . . . . . . 11 (((ω ∈ On ∧ 2𝑜 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑𝑜 2𝑜))
4543, 44mpan2 703 . . . . . . . . . 10 ((ω ∈ On ∧ 2𝑜 ∈ On) → ∅ ∈ (ω ↑𝑜 2𝑜))
464, 42, 45mp2an 704 . . . . . . . . 9 ∅ ∈ (ω ↑𝑜 2𝑜)
47 eqid 2610 . . . . . . . . . 10 (𝑓 ∘ (( I ↾ ((ω ↑𝑜 2𝑜) ∖ {∅, (𝑓‘∅)})) ∪ {⟨∅, (𝑓‘∅)⟩, ⟨(𝑓‘∅), ∅⟩})) = (𝑓 ∘ (( I ↾ ((ω ↑𝑜 2𝑜) ∖ {∅, (𝑓‘∅)})) ∪ {⟨∅, (𝑓‘∅)⟩, ⟨(𝑓‘∅), ∅⟩}))
4847fveqf1o 6457 . . . . . . . . 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‘∅), ∅⟩}))‘∅) = ∅))
4946, 43, 48mp3an23 1408 . . . . . . . 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‘∅), ∅⟩}))‘∅) = ∅))
5049ad2antll 761 . . . . . . 7 ((𝐴 ∈ On ∧ (∀𝑥𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-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‘∅), ∅⟩}))‘∅) = ∅))
5150simpld 474 . . . . . 6 ((𝐴 ∈ On ∧ (∀𝑥𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-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)
5250simprd 478 . . . . . 6 ((𝐴 ∈ On ∧ (∀𝑥𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦)) ∧ 𝑓:(ω ↑𝑜 2𝑜)–1-1-onto→ω)) → ((𝑓 ∘ (( I ↾ ((ω ↑𝑜 2𝑜) ∖ {∅, (𝑓‘∅)})) ∪ {⟨∅, (𝑓‘∅)⟩, ⟨(𝑓‘∅), ∅⟩}))‘∅) = ∅)
5321, 37, 41, 51, 52infxpenc2lem3 8727 . . . . 5 ((𝐴 ∈ On ∧ (∀𝑥𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦)) ∧ 𝑓:(ω ↑𝑜 2𝑜)–1-1-onto→ω)) → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → (𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
5453ex 449 . . . 4 (𝐴 ∈ On → ((∀𝑥𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦)) ∧ 𝑓:(ω ↑𝑜 2𝑜)–1-1-onto→ω) → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → (𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)))
5554exlimdvv 1849 . . 3 (𝐴 ∈ On → (∃𝑛𝑓(∀𝑥𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦)) ∧ 𝑓:(ω ↑𝑜 2𝑜)–1-1-onto→ω) → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → (𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)))
5620, 55syl5bir 232 . 2 (𝐴 ∈ On → ((∃𝑛𝑥𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦)) ∧ ∃𝑓 𝑓:(ω ↑𝑜 2𝑜)–1-1-onto→ω) → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → (𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)))
571, 19, 56mp2and 711 1 (𝐴 ∈ On → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → (𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  {cpr 4127  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   I cid 4948   × cxp 5036  ◡ccnv 5037  ran crn 5039   ↾ cres 5040   ∘ ccom 5042  Oncon0 5640  suc csuc 5642  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  ωcom 6957  1𝑜c1o 7440  2𝑜c2o 7441   ·𝑜 comu 7445   ↑𝑜 coe 7446   ≈ cen 7838 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-cnf 8442  df-card 8648 This theorem is referenced by:  pwfseq  9365
 Copyright terms: Public domain W3C validator