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Theorem infxpenc2 9045
Description: Existence form of infxpenc 9041. A "uniform" or "canonical" version of infxpen 9037, 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 8767 . 2 (𝐴 ∈ On → ∃𝑛𝑥𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦)))
2 df-2o 7714 . . . . . . . 8 2𝑜 = suc 1𝑜
32oveq2i 6804 . . . . . . 7 (ω ↑𝑜 2𝑜) = (ω ↑𝑜 suc 1𝑜)
4 omelon 8707 . . . . . . . 8 ω ∈ On
5 1on 7720 . . . . . . . 8 1𝑜 ∈ On
6 oesuc 7761 . . . . . . . 8 ((ω ∈ On ∧ 1𝑜 ∈ On) → (ω ↑𝑜 suc 1𝑜) = ((ω ↑𝑜 1𝑜) ·𝑜 ω))
74, 5, 6mp2an 672 . . . . . . 7 (ω ↑𝑜 suc 1𝑜) = ((ω ↑𝑜 1𝑜) ·𝑜 ω)
8 oe1 7778 . . . . . . . . 9 (ω ∈ On → (ω ↑𝑜 1𝑜) = ω)
94, 8ax-mp 5 . . . . . . . 8 (ω ↑𝑜 1𝑜) = ω
109oveq1i 6803 . . . . . . 7 ((ω ↑𝑜 1𝑜) ·𝑜 ω) = (ω ·𝑜 ω)
113, 7, 103eqtri 2797 . . . . . 6 (ω ↑𝑜 2𝑜) = (ω ·𝑜 ω)
12 omxpen 8218 . . . . . . 7 ((ω ∈ On ∧ ω ∈ On) → (ω ·𝑜 ω) ≈ (ω × ω))
134, 4, 12mp2an 672 . . . . . 6 (ω ·𝑜 ω) ≈ (ω × ω)
1411, 13eqbrtri 4807 . . . . 5 (ω ↑𝑜 2𝑜) ≈ (ω × ω)
15 xpomen 9038 . . . . 5 (ω × ω) ≈ ω
1614, 15entri 8163 . . . 4 (ω ↑𝑜 2𝑜) ≈ ω
1716a1i 11 . . 3 (𝐴 ∈ On → (ω ↑𝑜 2𝑜) ≈ ω)
18 bren 8118 . . 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 2344 . . 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 468 . . . . . 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 754 . . . . . . 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 3776 . . . . . . . . 9 (𝑥 = 𝑏 → (ω ⊆ 𝑥 ↔ ω ⊆ 𝑏))
24 oveq2 6801 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (ω ↑𝑜 𝑦) = (ω ↑𝑜 𝑤))
25 f1oeq3 6270 . . . . . . . . . . . 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 3321 . . . . . . . . . 10 (∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦) ↔ ∃𝑤 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑤))
28 fveq2 6332 . . . . . . . . . . . . 13 (𝑥 = 𝑏 → (𝑛𝑥) = (𝑛𝑏))
29 f1oeq1 6268 . . . . . . . . . . . . 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 6269 . . . . . . . . . . . 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 3200 . . . . . . . . . 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 3320 . . . . . . 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 6801 . . . . . . . . 9 (𝑏 = 𝑧 → (ω ↑𝑜 𝑏) = (ω ↑𝑜 𝑧))
3938cbvmptv 4884 . . . . . . . 8 (𝑏 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑏)) = (𝑧 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑧))
4039cnveqi 5435 . . . . . . 7 (𝑏 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑏)) = (𝑧 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑧))
4140fveq1i 6333 . . . . . 6 ((𝑏 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑏))‘ran (𝑛𝑏)) = ((𝑧 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑧))‘ran (𝑛𝑏))
42 2on 7722 . . . . . . . . . 10 2𝑜 ∈ On
43 peano1 7232 . . . . . . . . . . 11 ∅ ∈ ω
44 oen0 7820 . . . . . . . . . . 11 (((ω ∈ On ∧ 2𝑜 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑𝑜 2𝑜))
4543, 44mpan2 671 . . . . . . . . . 10 ((ω ∈ On ∧ 2𝑜 ∈ On) → ∅ ∈ (ω ↑𝑜 2𝑜))
464, 42, 45mp2an 672 . . . . . . . . 9 ∅ ∈ (ω ↑𝑜 2𝑜)
47 eqid 2771 . . . . . . . . . 10 (𝑓 ∘ (( I ↾ ((ω ↑𝑜 2𝑜) ∖ {∅, (𝑓‘∅)})) ∪ {⟨∅, (𝑓‘∅)⟩, ⟨(𝑓‘∅), ∅⟩})) = (𝑓 ∘ (( I ↾ ((ω ↑𝑜 2𝑜) ∖ {∅, (𝑓‘∅)})) ∪ {⟨∅, (𝑓‘∅)⟩, ⟨(𝑓‘∅), ∅⟩}))
4847fveqf1o 6700 . . . . . . . . 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 1564 . . . . . . . 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 708 . . . . . . 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 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‘∅), ∅⟩})):(ω ↑𝑜 2𝑜)–1-1-onto→ω)
5250simprd 483 . . . . . 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 9044 . . . . 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 397 . . . 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 2014 . . 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 679 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 382   = wceq 1631  wex 1852  wcel 2145  wral 3061  wrex 3062  cdif 3720  cun 3721  wss 3723  c0 4063  {cpr 4318  cop 4322   class class class wbr 4786  cmpt 4863   I cid 5156   × cxp 5247  ccnv 5248  ran crn 5250  cres 5251  ccom 5253  Oncon0 5866  suc csuc 5868  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6793  ωcom 7212  1𝑜c1o 7706  2𝑜c2o 7707   ·𝑜 comu 7711  𝑜 coe 7712  cen 8106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-supp 7447  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-seqom 7696  df-1o 7713  df-2o 7714  df-oadd 7717  df-omul 7718  df-oexp 7719  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fsupp 8432  df-oi 8571  df-cnf 8723  df-card 8965
This theorem is referenced by:  pwfseq  9688
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