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Theorem infxpenc2lem3 10062
Description: Lemma for infxpenc2 10063. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.)
Hypotheses
Ref Expression
infxpenc2.1 (𝜑𝐴 ∈ On)
infxpenc2.2 (𝜑 → ∀𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
infxpenc2.3 𝑊 = ((𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛𝑏))
infxpenc2.4 (𝜑𝐹:(ω ↑o 2o)–1-1-onto→ω)
infxpenc2.5 (𝜑 → (𝐹‘∅) = ∅)
Assertion
Ref Expression
infxpenc2lem3 (𝜑 → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → (𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
Distinct variable groups:   𝑔,𝑏,𝑛,𝑤,𝑥,𝐴   𝜑,𝑏,𝑤,𝑥   𝑔,𝑊,𝑤,𝑥   𝑔,𝐹,𝑥
Allowed substitution hints:   𝜑(𝑔,𝑛)   𝐹(𝑤,𝑛,𝑏)   𝑊(𝑛,𝑏)

Proof of Theorem infxpenc2lem3
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 infxpenc2.1 . 2 (𝜑𝐴 ∈ On)
2 infxpenc2.2 . 2 (𝜑 → ∀𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
3 infxpenc2.3 . 2 𝑊 = ((𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛𝑏))
4 infxpenc2.4 . 2 (𝜑𝐹:(ω ↑o 2o)–1-1-onto→ω)
5 infxpenc2.5 . 2 (𝜑 → (𝐹‘∅) = ∅)
6 eqid 2736 . 2 (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦( I ↾ 𝑊)))) = (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦( I ↾ 𝑊))))
7 eqid 2736 . 2 (((ω CNF 𝑊) ∘ (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦( I ↾ 𝑊))))) ∘ ((ω ↑o 2o) CNF 𝑊)) = (((ω CNF 𝑊) ∘ (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦( I ↾ 𝑊))))) ∘ ((ω ↑o 2o) CNF 𝑊))
8 eqid 2736 . 2 (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦((𝑧 ∈ 2o, 𝑤𝑊 ↦ ((2o ·o 𝑤) +o 𝑧)) ∘ (𝑧 ∈ 2o, 𝑤𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤)))))) = (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦((𝑧 ∈ 2o, 𝑤𝑊 ↦ ((2o ·o 𝑤) +o 𝑧)) ∘ (𝑧 ∈ 2o, 𝑤𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))))))
9 eqid 2736 . 2 (𝑧 ∈ 2o, 𝑤𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤)) = (𝑧 ∈ 2o, 𝑤𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))
10 eqid 2736 . 2 (𝑧 ∈ 2o, 𝑤𝑊 ↦ ((2o ·o 𝑤) +o 𝑧)) = (𝑧 ∈ 2o, 𝑤𝑊 ↦ ((2o ·o 𝑤) +o 𝑧))
11 eqid 2736 . 2 (((ω CNF (2o ·o 𝑊)) ∘ (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦((𝑧 ∈ 2o, 𝑤𝑊 ↦ ((2o ·o 𝑤) +o 𝑧)) ∘ (𝑧 ∈ 2o, 𝑤𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))))))) ∘ (ω CNF (𝑊 ·o 2o))) = (((ω CNF (2o ·o 𝑊)) ∘ (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦((𝑧 ∈ 2o, 𝑤𝑊 ↦ ((2o ·o 𝑤) +o 𝑧)) ∘ (𝑧 ∈ 2o, 𝑤𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))))))) ∘ (ω CNF (𝑊 ·o 2o)))
12 eqid 2736 . 2 (𝑥 ∈ (ω ↑o 𝑊), 𝑦 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑥) +o 𝑦)) = (𝑥 ∈ (ω ↑o 𝑊), 𝑦 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑥) +o 𝑦))
13 eqid 2736 . 2 (𝑥𝑏, 𝑦𝑏 ↦ ⟨((𝑛𝑏)‘𝑥), ((𝑛𝑏)‘𝑦)⟩) = (𝑥𝑏, 𝑦𝑏 ↦ ⟨((𝑛𝑏)‘𝑥), ((𝑛𝑏)‘𝑦)⟩)
14 eqid 2736 . 2 ((𝑛𝑏) ∘ ((((((ω CNF 𝑊) ∘ (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦( I ↾ 𝑊))))) ∘ ((ω ↑o 2o) CNF 𝑊)) ∘ (((ω CNF (2o ·o 𝑊)) ∘ (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦((𝑧 ∈ 2o, 𝑤𝑊 ↦ ((2o ·o 𝑤) +o 𝑧)) ∘ (𝑧 ∈ 2o, 𝑤𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))))))) ∘ (ω CNF (𝑊 ·o 2o)))) ∘ (𝑥 ∈ (ω ↑o 𝑊), 𝑦 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑥) +o 𝑦))) ∘ (𝑥𝑏, 𝑦𝑏 ↦ ⟨((𝑛𝑏)‘𝑥), ((𝑛𝑏)‘𝑦)⟩))) = ((𝑛𝑏) ∘ ((((((ω CNF 𝑊) ∘ (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦( I ↾ 𝑊))))) ∘ ((ω ↑o 2o) CNF 𝑊)) ∘ (((ω CNF (2o ·o 𝑊)) ∘ (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦((𝑧 ∈ 2o, 𝑤𝑊 ↦ ((2o ·o 𝑤) +o 𝑧)) ∘ (𝑧 ∈ 2o, 𝑤𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))))))) ∘ (ω CNF (𝑊 ·o 2o)))) ∘ (𝑥 ∈ (ω ↑o 𝑊), 𝑦 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑥) +o 𝑦))) ∘ (𝑥𝑏, 𝑦𝑏 ↦ ⟨((𝑛𝑏)‘𝑥), ((𝑛𝑏)‘𝑦)⟩)))
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14infxpenc2lem2 10061 1 (𝜑 → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → (𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wex 1778  wcel 2107  wral 3060  wrex 3069  {crab 3435  cdif 3947  wss 3950  c0 4332  cop 4631   class class class wbr 5142  cmpt 5224   I cid 5576   × cxp 5682  ccnv 5683  ran crn 5685  cres 5686  ccom 5688  Oncon0 6383  1-1-ontowf1o 6559  cfv 6560  (class class class)co 7432  cmpo 7434  ωcom 7888  1oc1o 8500  2oc2o 8501   +o coa 8504   ·o comu 8505  o coe 8506  m cmap 8867   finSupp cfsupp 9402   CNF ccnf 9702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-seqom 8489  df-1o 8507  df-2o 8508  df-oadd 8511  df-omul 8512  df-oexp 8513  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-oi 9551  df-cnf 9703
This theorem is referenced by:  infxpenc2  10063
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