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Theorem infxpenc2 10062
Description: Existence form of infxpenc 10058. A "uniform" or "canonical" version of infxpen 10054, 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 9746 . 2 (𝐴 ∈ On → ∃𝑛𝑥𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛𝑥):𝑥1-1-onto→(ω ↑o 𝑦)))
2 df-2o 8507 . . . . . . . 8 2o = suc 1o
32oveq2i 7442 . . . . . . 7 (ω ↑o 2o) = (ω ↑o suc 1o)
4 omelon 9686 . . . . . . . 8 ω ∈ On
5 1on 8518 . . . . . . . 8 1o ∈ On
6 oesuc 8565 . . . . . . . 8 ((ω ∈ On ∧ 1o ∈ On) → (ω ↑o suc 1o) = ((ω ↑o 1o) ·o ω))
74, 5, 6mp2an 692 . . . . . . 7 (ω ↑o suc 1o) = ((ω ↑o 1o) ·o ω)
8 oe1 8582 . . . . . . . . 9 (ω ∈ On → (ω ↑o 1o) = ω)
94, 8ax-mp 5 . . . . . . . 8 (ω ↑o 1o) = ω
109oveq1i 7441 . . . . . . 7 ((ω ↑o 1o) ·o ω) = (ω ·o ω)
113, 7, 103eqtri 2769 . . . . . 6 (ω ↑o 2o) = (ω ·o ω)
12 omxpen 9114 . . . . . . 7 ((ω ∈ On ∧ ω ∈ On) → (ω ·o ω) ≈ (ω × ω))
134, 4, 12mp2an 692 . . . . . 6 (ω ·o ω) ≈ (ω × ω)
1411, 13eqbrtri 5164 . . . . 5 (ω ↑o 2o) ≈ (ω × ω)
15 xpomen 10055 . . . . 5 (ω × ω) ≈ ω
1614, 15entri 9048 . . . 4 (ω ↑o 2o) ≈ ω
1716a1i 11 . . 3 (𝐴 ∈ On → (ω ↑o 2o) ≈ ω)
18 bren 8995 . . 3 ((ω ↑o 2o) ≈ ω ↔ ∃𝑓 𝑓:(ω ↑o 2o)–1-1-onto→ω)
1917, 18sylib 218 . 2 (𝐴 ∈ On → ∃𝑓 𝑓:(ω ↑o 2o)–1-1-onto→ω)
20 exdistrv 1955 . . 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 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)) → 𝐴 ∈ On)
22 simprl 771 . . . . . . 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 4010 . . . . . . . . 9 (𝑥 = 𝑏 → (ω ⊆ 𝑥 ↔ ω ⊆ 𝑏))
24 oveq2 7439 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (ω ↑o 𝑦) = (ω ↑o 𝑤))
2524f1oeq3d 6845 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝑛𝑥):𝑥1-1-onto→(ω ↑o 𝑦) ↔ (𝑛𝑥):𝑥1-1-onto→(ω ↑o 𝑤)))
2625cbvrexvw 3238 . . . . . . . . . 10 (∃𝑦 ∈ (On ∖ 1o)(𝑛𝑥):𝑥1-1-onto→(ω ↑o 𝑦) ↔ ∃𝑤 ∈ (On ∖ 1o)(𝑛𝑥):𝑥1-1-onto→(ω ↑o 𝑤))
27 fveq2 6906 . . . . . . . . . . . . 13 (𝑥 = 𝑏 → (𝑛𝑥) = (𝑛𝑏))
2827f1oeq1d 6843 . . . . . . . . . . . 12 (𝑥 = 𝑏 → ((𝑛𝑥):𝑥1-1-onto→(ω ↑o 𝑤) ↔ (𝑛𝑏):𝑥1-1-onto→(ω ↑o 𝑤)))
29 f1oeq2 6837 . . . . . . . . . . . 12 (𝑥 = 𝑏 → ((𝑛𝑏):𝑥1-1-onto→(ω ↑o 𝑤) ↔ (𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
3028, 29bitrd 279 . . . . . . . . . . 11 (𝑥 = 𝑏 → ((𝑛𝑥):𝑥1-1-onto→(ω ↑o 𝑤) ↔ (𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
3130rexbidv 3179 . . . . . . . . . 10 (𝑥 = 𝑏 → (∃𝑤 ∈ (On ∖ 1o)(𝑛𝑥):𝑥1-1-onto→(ω ↑o 𝑤) ↔ ∃𝑤 ∈ (On ∖ 1o)(𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
3226, 31bitrid 283 . . . . . . . . 9 (𝑥 = 𝑏 → (∃𝑦 ∈ (On ∖ 1o)(𝑛𝑥):𝑥1-1-onto→(ω ↑o 𝑦) ↔ ∃𝑤 ∈ (On ∖ 1o)(𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
3323, 32imbi12d 344 . . . . . . . 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))))
3433cbvralvw 3237 . . . . . . 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)))
3522, 34sylib 218 . . . . . 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)))
36 oveq2 7439 . . . . . . . . 9 (𝑏 = 𝑧 → (ω ↑o 𝑏) = (ω ↑o 𝑧))
3736cbvmptv 5255 . . . . . . . 8 (𝑏 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑏)) = (𝑧 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑧))
3837cnveqi 5885 . . . . . . 7 (𝑏 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑏)) = (𝑧 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑧))
3938fveq1i 6907 . . . . . 6 ((𝑏 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑏))‘ran (𝑛𝑏)) = ((𝑧 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑧))‘ran (𝑛𝑏))
40 2on 8520 . . . . . . . . . 10 2o ∈ On
41 peano1 7910 . . . . . . . . . . 11 ∅ ∈ ω
42 oen0 8624 . . . . . . . . . . 11 (((ω ∈ On ∧ 2o ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 2o))
4341, 42mpan2 691 . . . . . . . . . 10 ((ω ∈ On ∧ 2o ∈ On) → ∅ ∈ (ω ↑o 2o))
444, 40, 43mp2an 692 . . . . . . . . 9 ∅ ∈ (ω ↑o 2o)
45 eqid 2737 . . . . . . . . . 10 (𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (𝑓‘∅)})) ∪ {⟨∅, (𝑓‘∅)⟩, ⟨(𝑓‘∅), ∅⟩})) = (𝑓 ∘ (( I ↾ ((ω ↑o 2o) ∖ {∅, (𝑓‘∅)})) ∪ {⟨∅, (𝑓‘∅)⟩, ⟨(𝑓‘∅), ∅⟩}))
4645fveqf1o 7322 . . . . . . . . 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‘∅), ∅⟩}))‘∅) = ∅))
4744, 41, 46mp3an23 1455 . . . . . . . 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‘∅), ∅⟩}))‘∅) = ∅))
4847ad2antll 729 . . . . . . 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‘∅), ∅⟩}))‘∅) = ∅))
4948simpld 494 . . . . . 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)
5048simprd 495 . . . . . 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‘∅), ∅⟩}))‘∅) = ∅)
5121, 35, 39, 49, 50infxpenc2lem3 10061 . . . . 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))
5251ex 412 . . . 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)))
5352exlimdvv 1934 . . 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)))
5420, 53biimtrrid 243 . 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)))
551, 19, 54mp2and 699 1 (𝐴 ∈ On → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → (𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  wral 3061  wrex 3070  cdif 3948  cun 3949  wss 3951  c0 4333  {cpr 4628  cop 4632   class class class wbr 5143  cmpt 5225   I cid 5577   × cxp 5683  ccnv 5684  ran crn 5686  cres 5687  ccom 5689  Oncon0 6384  suc csuc 6386  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  ωcom 7887  1oc1o 8499  2oc2o 8500   ·o comu 8504  o coe 8505  cen 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seqom 8488  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-oexp 8512  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-oi 9550  df-cnf 9702  df-card 9979
This theorem is referenced by:  pwfseq  10704
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