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Theorem infxpenc2 10060
Description: Existence form of infxpenc 10056. A "uniform" or "canonical" version of infxpen 10052, 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 9744 . 2 (𝐴 ∈ On → ∃𝑛𝑥𝐴 (ω ⊆ 𝑥 → ∃𝑦 ∈ (On ∖ 1o)(𝑛𝑥):𝑥1-1-onto→(ω ↑o 𝑦)))
2 df-2o 8506 . . . . . . . 8 2o = suc 1o
32oveq2i 7442 . . . . . . 7 (ω ↑o 2o) = (ω ↑o suc 1o)
4 omelon 9684 . . . . . . . 8 ω ∈ On
5 1on 8517 . . . . . . . 8 1o ∈ On
6 oesuc 8564 . . . . . . . 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 8581 . . . . . . . . 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 2767 . . . . . 6 (ω ↑o 2o) = (ω ·o ω)
12 omxpen 9113 . . . . . . 7 ((ω ∈ On ∧ ω ∈ On) → (ω ·o ω) ≈ (ω × ω))
134, 4, 12mp2an 692 . . . . . 6 (ω ·o ω) ≈ (ω × ω)
1411, 13eqbrtri 5169 . . . . 5 (ω ↑o 2o) ≈ (ω × ω)
15 xpomen 10053 . . . . 5 (ω × ω) ≈ ω
1614, 15entri 9047 . . . 4 (ω ↑o 2o) ≈ ω
1716a1i 11 . . 3 (𝐴 ∈ On → (ω ↑o 2o) ≈ ω)
18 bren 8994 . . 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 1953 . . 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 4022 . . . . . . . . 9 (𝑥 = 𝑏 → (ω ⊆ 𝑥 ↔ ω ⊆ 𝑏))
24 oveq2 7439 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (ω ↑o 𝑦) = (ω ↑o 𝑤))
2524f1oeq3d 6846 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝑛𝑥):𝑥1-1-onto→(ω ↑o 𝑦) ↔ (𝑛𝑥):𝑥1-1-onto→(ω ↑o 𝑤)))
2625cbvrexvw 3236 . . . . . . . . . 10 (∃𝑦 ∈ (On ∖ 1o)(𝑛𝑥):𝑥1-1-onto→(ω ↑o 𝑦) ↔ ∃𝑤 ∈ (On ∖ 1o)(𝑛𝑥):𝑥1-1-onto→(ω ↑o 𝑤))
27 fveq2 6907 . . . . . . . . . . . . 13 (𝑥 = 𝑏 → (𝑛𝑥) = (𝑛𝑏))
2827f1oeq1d 6844 . . . . . . . . . . . 12 (𝑥 = 𝑏 → ((𝑛𝑥):𝑥1-1-onto→(ω ↑o 𝑤) ↔ (𝑛𝑏):𝑥1-1-onto→(ω ↑o 𝑤)))
29 f1oeq2 6838 . . . . . . . . . . . 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 3177 . . . . . . . . . 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 3235 . . . . . . 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 5261 . . . . . . . 8 (𝑏 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑏)) = (𝑧 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑧))
3837cnveqi 5888 . . . . . . 7 (𝑏 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑏)) = (𝑧 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑧))
3938fveq1i 6908 . . . . . 6 ((𝑏 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑏))‘ran (𝑛𝑏)) = ((𝑧 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑧))‘ran (𝑛𝑏))
40 2on 8519 . . . . . . . . . 10 2o ∈ On
41 peano1 7911 . . . . . . . . . . 11 ∅ ∈ ω
42 oen0 8623 . . . . . . . . . . 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 2735 . . . . . . . . . 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 1452 . . . . . . . 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 10059 . . . . 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 1932 . . 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 1537  wex 1776  wcel 2106  wral 3059  wrex 3068  cdif 3960  cun 3961  wss 3963  c0 4339  {cpr 4633  cop 4637   class class class wbr 5148  cmpt 5231   I cid 5582   × cxp 5687  ccnv 5688  ran crn 5690  cres 5691  ccom 5693  Oncon0 6386  suc csuc 6388  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  ωcom 7887  1oc1o 8498  2oc2o 8499   ·o comu 8503  o coe 8504  cen 8981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-seqom 8487  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510  df-oexp 8511  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-oi 9548  df-cnf 9700  df-card 9977
This theorem is referenced by:  pwfseq  10702
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