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

Theorem infxpenc2 9055
Description: Existence form of infxpenc 9051. A "uniform" or "canonical" version of infxpen 9047, 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 8778 . 2 (𝐴 ∈ On → ∃𝑛𝑥𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦)))
2 df-2o 7731 . . . . . . . 8 2𝑜 = suc 1𝑜
32oveq2i 6825 . . . . . . 7 (ω ↑𝑜 2𝑜) = (ω ↑𝑜 suc 1𝑜)
4 omelon 8718 . . . . . . . 8 ω ∈ On
5 1on 7737 . . . . . . . 8 1𝑜 ∈ On
6 oesuc 7778 . . . . . . . 8 ((ω ∈ On ∧ 1𝑜 ∈ On) → (ω ↑𝑜 suc 1𝑜) = ((ω ↑𝑜 1𝑜) ·𝑜 ω))
74, 5, 6mp2an 710 . . . . . . 7 (ω ↑𝑜 suc 1𝑜) = ((ω ↑𝑜 1𝑜) ·𝑜 ω)
8 oe1 7795 . . . . . . . . 9 (ω ∈ On → (ω ↑𝑜 1𝑜) = ω)
94, 8ax-mp 5 . . . . . . . 8 (ω ↑𝑜 1𝑜) = ω
109oveq1i 6824 . . . . . . 7 ((ω ↑𝑜 1𝑜) ·𝑜 ω) = (ω ·𝑜 ω)
113, 7, 103eqtri 2786 . . . . . 6 (ω ↑𝑜 2𝑜) = (ω ·𝑜 ω)
12 omxpen 8229 . . . . . . 7 ((ω ∈ On ∧ ω ∈ On) → (ω ·𝑜 ω) ≈ (ω × ω))
134, 4, 12mp2an 710 . . . . . 6 (ω ·𝑜 ω) ≈ (ω × ω)
1411, 13eqbrtri 4825 . . . . 5 (ω ↑𝑜 2𝑜) ≈ (ω × ω)
15 xpomen 9048 . . . . 5 (ω × ω) ≈ ω
1614, 15entri 8177 . . . 4 (ω ↑𝑜 2𝑜) ≈ ω
1716a1i 11 . . 3 (𝐴 ∈ On → (ω ↑𝑜 2𝑜) ≈ ω)
18 bren 8132 . . 3 ((ω ↑𝑜 2𝑜) ≈ ω ↔ ∃𝑓 𝑓:(ω ↑𝑜 2𝑜)–1-1-onto→ω)
1917, 18sylib 208 . 2 (𝐴 ∈ On → ∃𝑓 𝑓:(ω ↑𝑜 2𝑜)–1-1-onto→ω)
20 eeanv 2327 . . 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 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)) → 𝐴 ∈ On)
22 simprl 811 . . . . . . 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 3768 . . . . . . . . 9 (𝑥 = 𝑏 → (ω ⊆ 𝑥 ↔ ω ⊆ 𝑏))
24 oveq2 6822 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (ω ↑𝑜 𝑦) = (ω ↑𝑜 𝑤))
25 f1oeq3 6291 . . . . . . . . . . . 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 3311 . . . . . . . . . 10 (∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦) ↔ ∃𝑤 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑤))
28 fveq2 6353 . . . . . . . . . . . . 13 (𝑥 = 𝑏 → (𝑛𝑥) = (𝑛𝑏))
29 f1oeq1 6289 . . . . . . . . . . . . 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 6290 . . . . . . . . . . . 12 (𝑥 = 𝑏 → ((𝑛𝑏):𝑥1-1-onto→(ω ↑𝑜 𝑤) ↔ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
3230, 31bitrd 268 . . . . . . . . . . 11 (𝑥 = 𝑏 → ((𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑤) ↔ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
3332rexbidv 3190 . . . . . . . . . 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 272 . . . . . . . . 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 3310 . . . . . . 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 208 . . . . . 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 6822 . . . . . . . . 9 (𝑏 = 𝑧 → (ω ↑𝑜 𝑏) = (ω ↑𝑜 𝑧))
3938cbvmptv 4902 . . . . . . . 8 (𝑏 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑏)) = (𝑧 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑧))
4039cnveqi 5452 . . . . . . 7 (𝑏 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑏)) = (𝑧 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑧))
4140fveq1i 6354 . . . . . 6 ((𝑏 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑏))‘ran (𝑛𝑏)) = ((𝑧 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑧))‘ran (𝑛𝑏))
42 2on 7739 . . . . . . . . . 10 2𝑜 ∈ On
43 peano1 7251 . . . . . . . . . . 11 ∅ ∈ ω
44 oen0 7837 . . . . . . . . . . 11 (((ω ∈ On ∧ 2𝑜 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑𝑜 2𝑜))
4543, 44mpan2 709 . . . . . . . . . 10 ((ω ∈ On ∧ 2𝑜 ∈ On) → ∅ ∈ (ω ↑𝑜 2𝑜))
464, 42, 45mp2an 710 . . . . . . . . 9 ∅ ∈ (ω ↑𝑜 2𝑜)
47 eqid 2760 . . . . . . . . . 10 (𝑓 ∘ (( I ↾ ((ω ↑𝑜 2𝑜) ∖ {∅, (𝑓‘∅)})) ∪ {⟨∅, (𝑓‘∅)⟩, ⟨(𝑓‘∅), ∅⟩})) = (𝑓 ∘ (( I ↾ ((ω ↑𝑜 2𝑜) ∖ {∅, (𝑓‘∅)})) ∪ {⟨∅, (𝑓‘∅)⟩, ⟨(𝑓‘∅), ∅⟩}))
4847fveqf1o 6721 . . . . . . . . 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 1565 . . . . . . . 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 767 . . . . . . 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 477 . . . . . 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 482 . . . . . 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 9054 . . . . 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 2011 . . 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 233 . 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 717 1 (𝐴 ∈ On → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → (𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wex 1853  wcel 2139  wral 3050  wrex 3051  cdif 3712  cun 3713  wss 3715  c0 4058  {cpr 4323  cop 4327   class class class wbr 4804  cmpt 4881   I cid 5173   × cxp 5264  ccnv 5265  ran crn 5267  cres 5268  ccom 5270  Oncon0 5884  suc csuc 5886  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6814  ωcom 7231  1𝑜c1o 7723  2𝑜c2o 7724   ·𝑜 comu 7728  𝑜 coe 7729  cen 8120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-supp 7465  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-seqom 7713  df-1o 7730  df-2o 7731  df-oadd 7734  df-omul 7735  df-oexp 7736  df-er 7913  df-map 8027  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-fsupp 8443  df-oi 8582  df-cnf 8734  df-card 8975
This theorem is referenced by:  pwfseq  9698
  Copyright terms: Public domain W3C validator