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Theorem infxpenc2 8830
Description: Existence form of infxpenc 8826. A "uniform" or "canonical" version of infxpen 8822, 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 8588 . 2 (𝐴 ∈ On → ∃𝑛𝑥𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦)))
2 df-2o 7546 . . . . . . . 8 2𝑜 = suc 1𝑜
32oveq2i 6646 . . . . . . 7 (ω ↑𝑜 2𝑜) = (ω ↑𝑜 suc 1𝑜)
4 omelon 8528 . . . . . . . 8 ω ∈ On
5 1on 7552 . . . . . . . 8 1𝑜 ∈ On
6 oesuc 7592 . . . . . . . 8 ((ω ∈ On ∧ 1𝑜 ∈ On) → (ω ↑𝑜 suc 1𝑜) = ((ω ↑𝑜 1𝑜) ·𝑜 ω))
74, 5, 6mp2an 707 . . . . . . 7 (ω ↑𝑜 suc 1𝑜) = ((ω ↑𝑜 1𝑜) ·𝑜 ω)
8 oe1 7609 . . . . . . . . 9 (ω ∈ On → (ω ↑𝑜 1𝑜) = ω)
94, 8ax-mp 5 . . . . . . . 8 (ω ↑𝑜 1𝑜) = ω
109oveq1i 6645 . . . . . . 7 ((ω ↑𝑜 1𝑜) ·𝑜 ω) = (ω ·𝑜 ω)
113, 7, 103eqtri 2646 . . . . . 6 (ω ↑𝑜 2𝑜) = (ω ·𝑜 ω)
12 omxpen 8047 . . . . . . 7 ((ω ∈ On ∧ ω ∈ On) → (ω ·𝑜 ω) ≈ (ω × ω))
134, 4, 12mp2an 707 . . . . . 6 (ω ·𝑜 ω) ≈ (ω × ω)
1411, 13eqbrtri 4665 . . . . 5 (ω ↑𝑜 2𝑜) ≈ (ω × ω)
15 xpomen 8823 . . . . 5 (ω × ω) ≈ ω
1614, 15entri 7995 . . . 4 (ω ↑𝑜 2𝑜) ≈ ω
1716a1i 11 . . 3 (𝐴 ∈ On → (ω ↑𝑜 2𝑜) ≈ ω)
18 bren 7949 . . 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 2180 . . 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 473 . . . . . 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 793 . . . . . . 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 3619 . . . . . . . . 9 (𝑥 = 𝑏 → (ω ⊆ 𝑥 ↔ ω ⊆ 𝑏))
24 oveq2 6643 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (ω ↑𝑜 𝑦) = (ω ↑𝑜 𝑤))
25 f1oeq3 6116 . . . . . . . . . . . 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 3167 . . . . . . . . . 10 (∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦) ↔ ∃𝑤 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑤))
28 fveq2 6178 . . . . . . . . . . . . 13 (𝑥 = 𝑏 → (𝑛𝑥) = (𝑛𝑏))
29 f1oeq1 6114 . . . . . . . . . . . . 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 6115 . . . . . . . . . . . 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 3048 . . . . . . . . . 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 334 . . . . . . . 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 3166 . . . . . . 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 6643 . . . . . . . . 9 (𝑏 = 𝑧 → (ω ↑𝑜 𝑏) = (ω ↑𝑜 𝑧))
3938cbvmptv 4741 . . . . . . . 8 (𝑏 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑏)) = (𝑧 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑧))
4039cnveqi 5286 . . . . . . 7 (𝑏 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑏)) = (𝑧 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑧))
4140fveq1i 6179 . . . . . 6 ((𝑏 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑏))‘ran (𝑛𝑏)) = ((𝑧 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑧))‘ran (𝑛𝑏))
42 2on 7553 . . . . . . . . . 10 2𝑜 ∈ On
43 peano1 7070 . . . . . . . . . . 11 ∅ ∈ ω
44 oen0 7651 . . . . . . . . . . 11 (((ω ∈ On ∧ 2𝑜 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑𝑜 2𝑜))
4543, 44mpan2 706 . . . . . . . . . 10 ((ω ∈ On ∧ 2𝑜 ∈ On) → ∅ ∈ (ω ↑𝑜 2𝑜))
464, 42, 45mp2an 707 . . . . . . . . 9 ∅ ∈ (ω ↑𝑜 2𝑜)
47 eqid 2620 . . . . . . . . . 10 (𝑓 ∘ (( I ↾ ((ω ↑𝑜 2𝑜) ∖ {∅, (𝑓‘∅)})) ∪ {⟨∅, (𝑓‘∅)⟩, ⟨(𝑓‘∅), ∅⟩})) = (𝑓 ∘ (( I ↾ ((ω ↑𝑜 2𝑜) ∖ {∅, (𝑓‘∅)})) ∪ {⟨∅, (𝑓‘∅)⟩, ⟨(𝑓‘∅), ∅⟩}))
4847fveqf1o 6542 . . . . . . . . 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 1414 . . . . . . . 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 764 . . . . . . 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 475 . . . . . 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 479 . . . . . 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 8829 . . . . 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 450 . . . 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 1860 . . 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 714 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 384   = wceq 1481  wex 1702  wcel 1988  wral 2909  wrex 2910  cdif 3564  cun 3565  wss 3567  c0 3907  {cpr 4170  cop 4174   class class class wbr 4644  cmpt 4720   I cid 5013   × cxp 5102  ccnv 5103  ran crn 5105  cres 5106  ccom 5108  Oncon0 5711  suc csuc 5713  1-1-ontowf1o 5875  cfv 5876  (class class class)co 6635  ωcom 7050  1𝑜c1o 7538  2𝑜c2o 7539   ·𝑜 comu 7543  𝑜 coe 7544  cen 7937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-supp 7281  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-seqom 7528  df-1o 7545  df-2o 7546  df-oadd 7549  df-omul 7550  df-oexp 7551  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-fsupp 8261  df-oi 8400  df-cnf 8544  df-card 8750
This theorem is referenced by:  pwfseq  9471
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