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Theorem infxpenc2 8789
Description: Existence form of infxpenc 8785. A "uniform" or "canonical" version of infxpen 8781, 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 8547 . 2 (𝐴 ∈ On → ∃𝑛𝑥𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦)))
2 df-2o 7506 . . . . . . . 8 2𝑜 = suc 1𝑜
32oveq2i 6615 . . . . . . 7 (ω ↑𝑜 2𝑜) = (ω ↑𝑜 suc 1𝑜)
4 omelon 8487 . . . . . . . 8 ω ∈ On
5 1on 7512 . . . . . . . 8 1𝑜 ∈ On
6 oesuc 7552 . . . . . . . 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 7569 . . . . . . . . 9 (ω ∈ On → (ω ↑𝑜 1𝑜) = ω)
94, 8ax-mp 5 . . . . . . . 8 (ω ↑𝑜 1𝑜) = ω
109oveq1i 6614 . . . . . . 7 ((ω ↑𝑜 1𝑜) ·𝑜 ω) = (ω ·𝑜 ω)
113, 7, 103eqtri 2647 . . . . . 6 (ω ↑𝑜 2𝑜) = (ω ·𝑜 ω)
12 omxpen 8006 . . . . . . 7 ((ω ∈ On ∧ ω ∈ On) → (ω ·𝑜 ω) ≈ (ω × ω))
134, 4, 12mp2an 707 . . . . . 6 (ω ·𝑜 ω) ≈ (ω × ω)
1411, 13eqbrtri 4634 . . . . 5 (ω ↑𝑜 2𝑜) ≈ (ω × ω)
15 xpomen 8782 . . . . 5 (ω × ω) ≈ ω
1614, 15entri 7954 . . . 4 (ω ↑𝑜 2𝑜) ≈ ω
1716a1i 11 . . 3 (𝐴 ∈ On → (ω ↑𝑜 2𝑜) ≈ ω)
18 bren 7908 . . 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 2181 . . 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 3606 . . . . . . . . 9 (𝑥 = 𝑏 → (ω ⊆ 𝑥 ↔ ω ⊆ 𝑏))
24 oveq2 6612 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (ω ↑𝑜 𝑦) = (ω ↑𝑜 𝑤))
25 f1oeq3 6086 . . . . . . . . . . . 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 3160 . . . . . . . . . 10 (∃𝑦 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑦) ↔ ∃𝑤 ∈ (On ∖ 1𝑜)(𝑛𝑥):𝑥1-1-onto→(ω ↑𝑜 𝑤))
28 fveq2 6148 . . . . . . . . . . . . 13 (𝑥 = 𝑏 → (𝑛𝑥) = (𝑛𝑏))
29 f1oeq1 6084 . . . . . . . . . . . . 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 6085 . . . . . . . . . . . 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 3045 . . . . . . . . . 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 3159 . . . . . . 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 6612 . . . . . . . . 9 (𝑏 = 𝑧 → (ω ↑𝑜 𝑏) = (ω ↑𝑜 𝑧))
3938cbvmptv 4710 . . . . . . . 8 (𝑏 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑏)) = (𝑧 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑧))
4039cnveqi 5257 . . . . . . 7 (𝑏 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑏)) = (𝑧 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑧))
4140fveq1i 6149 . . . . . 6 ((𝑏 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑏))‘ran (𝑛𝑏)) = ((𝑧 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑧))‘ran (𝑛𝑏))
42 2on 7513 . . . . . . . . . 10 2𝑜 ∈ On
43 peano1 7032 . . . . . . . . . . 11 ∅ ∈ ω
44 oen0 7611 . . . . . . . . . . 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 2621 . . . . . . . . . 10 (𝑓 ∘ (( I ↾ ((ω ↑𝑜 2𝑜) ∖ {∅, (𝑓‘∅)})) ∪ {⟨∅, (𝑓‘∅)⟩, ⟨(𝑓‘∅), ∅⟩})) = (𝑓 ∘ (( I ↾ ((ω ↑𝑜 2𝑜) ∖ {∅, (𝑓‘∅)})) ∪ {⟨∅, (𝑓‘∅)⟩, ⟨(𝑓‘∅), ∅⟩}))
4847fveqf1o 6511 . . . . . . . . 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 1413 . . . . . . . 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 8788 . . . . 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 1859 . . 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 1480  wex 1701  wcel 1987  wral 2907  wrex 2908  cdif 3552  cun 3553  wss 3555  c0 3891  {cpr 4150  cop 4154   class class class wbr 4613  cmpt 4673   I cid 4984   × cxp 5072  ccnv 5073  ran crn 5075  cres 5076  ccom 5078  Oncon0 5682  suc csuc 5684  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  ωcom 7012  1𝑜c1o 7498  2𝑜c2o 7499   ·𝑜 comu 7503  𝑜 coe 7504  cen 7896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-seqom 7488  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510  df-oexp 7511  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-oi 8359  df-cnf 8503  df-card 8709
This theorem is referenced by:  pwfseq  9430
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