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Theorem infxpenc2lem2 8726
Description: Lemma for infxpenc2 8728. (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‘∅) = ∅)
infxpenc2.k 𝐾 = (𝑦 ∈ {𝑥 ∈ ((ω ↑𝑜 2𝑜) ↑𝑚 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦( I ↾ 𝑊))))
infxpenc2.h 𝐻 = (((ω CNF 𝑊) ∘ 𝐾) ∘ ((ω ↑𝑜 2𝑜) CNF 𝑊))
infxpenc2.l 𝐿 = (𝑦 ∈ {𝑥 ∈ (ω ↑𝑚 (𝑊 ·𝑜 2𝑜)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦(𝑌𝑋))))
infxpenc2.x 𝑋 = (𝑧 ∈ 2𝑜, 𝑤𝑊 ↦ ((𝑊 ·𝑜 𝑧) +𝑜 𝑤))
infxpenc2.y 𝑌 = (𝑧 ∈ 2𝑜, 𝑤𝑊 ↦ ((2𝑜 ·𝑜 𝑤) +𝑜 𝑧))
infxpenc2.j 𝐽 = (((ω CNF (2𝑜 ·𝑜 𝑊)) ∘ 𝐿) ∘ (ω CNF (𝑊 ·𝑜 2𝑜)))
infxpenc2.z 𝑍 = (𝑥 ∈ (ω ↑𝑜 𝑊), 𝑦 ∈ (ω ↑𝑜 𝑊) ↦ (((ω ↑𝑜 𝑊) ·𝑜 𝑥) +𝑜 𝑦))
infxpenc2.t 𝑇 = (𝑥𝑏, 𝑦𝑏 ↦ ⟨((𝑛𝑏)‘𝑥), ((𝑛𝑏)‘𝑦)⟩)
infxpenc2.g 𝐺 = ((𝑛𝑏) ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇))
Assertion
Ref Expression
infxpenc2lem2 (𝜑 → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → (𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
Distinct variable groups:   𝑔,𝑏,𝑛,𝑤,𝑥,𝑦,𝐴   𝜑,𝑏,𝑤,𝑥,𝑦   𝑧,𝑔,𝑊,𝑤,𝑥,𝑦   𝑔,𝐹,𝑥,𝑦   𝑔,𝐺   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝜑(𝑧,𝑔,𝑛)   𝐴(𝑧)   𝑇(𝑥,𝑦,𝑧,𝑤,𝑔,𝑛,𝑏)   𝐹(𝑧,𝑤,𝑛,𝑏)   𝐺(𝑥,𝑦,𝑧,𝑤,𝑛,𝑏)   𝐻(𝑥,𝑦,𝑧,𝑤,𝑔,𝑛,𝑏)   𝐽(𝑥,𝑦,𝑧,𝑤,𝑔,𝑛,𝑏)   𝐾(𝑥,𝑦,𝑧,𝑤,𝑔,𝑛,𝑏)   𝐿(𝑥,𝑦,𝑧,𝑤,𝑔,𝑛,𝑏)   𝑊(𝑛,𝑏)   𝑋(𝑧,𝑤,𝑔,𝑛,𝑏)   𝑌(𝑧,𝑤,𝑔,𝑛,𝑏)   𝑍(𝑥,𝑦,𝑧,𝑤,𝑔,𝑛,𝑏)

Proof of Theorem infxpenc2lem2
StepHypRef Expression
1 infxpenc2.1 . . 3 (𝜑𝐴 ∈ On)
2 mptexg 6389 . . 3 (𝐴 ∈ On → (𝑏𝐴𝐺) ∈ V)
31, 2syl 17 . 2 (𝜑 → (𝑏𝐴𝐺) ∈ V)
41adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → 𝐴 ∈ On)
5 simprl 790 . . . . . . 7 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → 𝑏𝐴)
6 onelon 5665 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑏𝐴) → 𝑏 ∈ On)
74, 5, 6syl2anc 691 . . . . . 6 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → 𝑏 ∈ On)
8 simprr 792 . . . . . 6 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → ω ⊆ 𝑏)
9 infxpenc2.2 . . . . . . . 8 (𝜑 → ∀𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
10 infxpenc2.3 . . . . . . . 8 𝑊 = ((𝑥 ∈ (On ∖ 1𝑜) ↦ (ω ↑𝑜 𝑥))‘ran (𝑛𝑏))
111, 9, 10infxpenc2lem1 8725 . . . . . . 7 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → (𝑊 ∈ (On ∖ 1𝑜) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑊)))
1211simpld 474 . . . . . 6 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → 𝑊 ∈ (On ∖ 1𝑜))
13 infxpenc2.4 . . . . . . 7 (𝜑𝐹:(ω ↑𝑜 2𝑜)–1-1-onto→ω)
1413adantr 480 . . . . . 6 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → 𝐹:(ω ↑𝑜 2𝑜)–1-1-onto→ω)
15 infxpenc2.5 . . . . . . 7 (𝜑 → (𝐹‘∅) = ∅)
1615adantr 480 . . . . . 6 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → (𝐹‘∅) = ∅)
1711simprd 478 . . . . . 6 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → (𝑛𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑊))
18 infxpenc2.k . . . . . 6 𝐾 = (𝑦 ∈ {𝑥 ∈ ((ω ↑𝑜 2𝑜) ↑𝑚 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦( I ↾ 𝑊))))
19 infxpenc2.h . . . . . 6 𝐻 = (((ω CNF 𝑊) ∘ 𝐾) ∘ ((ω ↑𝑜 2𝑜) CNF 𝑊))
20 infxpenc2.l . . . . . 6 𝐿 = (𝑦 ∈ {𝑥 ∈ (ω ↑𝑚 (𝑊 ·𝑜 2𝑜)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦(𝑌𝑋))))
21 infxpenc2.x . . . . . 6 𝑋 = (𝑧 ∈ 2𝑜, 𝑤𝑊 ↦ ((𝑊 ·𝑜 𝑧) +𝑜 𝑤))
22 infxpenc2.y . . . . . 6 𝑌 = (𝑧 ∈ 2𝑜, 𝑤𝑊 ↦ ((2𝑜 ·𝑜 𝑤) +𝑜 𝑧))
23 infxpenc2.j . . . . . 6 𝐽 = (((ω CNF (2𝑜 ·𝑜 𝑊)) ∘ 𝐿) ∘ (ω CNF (𝑊 ·𝑜 2𝑜)))
24 infxpenc2.z . . . . . 6 𝑍 = (𝑥 ∈ (ω ↑𝑜 𝑊), 𝑦 ∈ (ω ↑𝑜 𝑊) ↦ (((ω ↑𝑜 𝑊) ·𝑜 𝑥) +𝑜 𝑦))
25 infxpenc2.t . . . . . 6 𝑇 = (𝑥𝑏, 𝑦𝑏 ↦ ⟨((𝑛𝑏)‘𝑥), ((𝑛𝑏)‘𝑦)⟩)
26 infxpenc2.g . . . . . 6 𝐺 = ((𝑛𝑏) ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇))
277, 8, 12, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26infxpenc 8724 . . . . 5 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → 𝐺:(𝑏 × 𝑏)–1-1-onto𝑏)
28 f1of 6050 . . . . . . . . 9 (𝐺:(𝑏 × 𝑏)–1-1-onto𝑏𝐺:(𝑏 × 𝑏)⟶𝑏)
2927, 28syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → 𝐺:(𝑏 × 𝑏)⟶𝑏)
30 vex 3176 . . . . . . . . 9 𝑏 ∈ V
3130, 30xpex 6860 . . . . . . . 8 (𝑏 × 𝑏) ∈ V
32 fex 6394 . . . . . . . 8 ((𝐺:(𝑏 × 𝑏)⟶𝑏 ∧ (𝑏 × 𝑏) ∈ V) → 𝐺 ∈ V)
3329, 31, 32sylancl 693 . . . . . . 7 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → 𝐺 ∈ V)
34 eqid 2610 . . . . . . . 8 (𝑏𝐴𝐺) = (𝑏𝐴𝐺)
3534fvmpt2 6200 . . . . . . 7 ((𝑏𝐴𝐺 ∈ V) → ((𝑏𝐴𝐺)‘𝑏) = 𝐺)
365, 33, 35syl2anc 691 . . . . . 6 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → ((𝑏𝐴𝐺)‘𝑏) = 𝐺)
37 f1oeq1 6040 . . . . . 6 (((𝑏𝐴𝐺)‘𝑏) = 𝐺 → (((𝑏𝐴𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto𝑏𝐺:(𝑏 × 𝑏)–1-1-onto𝑏))
3836, 37syl 17 . . . . 5 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → (((𝑏𝐴𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto𝑏𝐺:(𝑏 × 𝑏)–1-1-onto𝑏))
3927, 38mpbird 246 . . . 4 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → ((𝑏𝐴𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)
4039expr 641 . . 3 ((𝜑𝑏𝐴) → (ω ⊆ 𝑏 → ((𝑏𝐴𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
4140ralrimiva 2949 . 2 (𝜑 → ∀𝑏𝐴 (ω ⊆ 𝑏 → ((𝑏𝐴𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
42 nfmpt1 4675 . . . . 5 𝑏(𝑏𝐴𝐺)
4342nfeq2 2766 . . . 4 𝑏 𝑔 = (𝑏𝐴𝐺)
44 fveq1 6102 . . . . . 6 (𝑔 = (𝑏𝐴𝐺) → (𝑔𝑏) = ((𝑏𝐴𝐺)‘𝑏))
45 f1oeq1 6040 . . . . . 6 ((𝑔𝑏) = ((𝑏𝐴𝐺)‘𝑏) → ((𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ ((𝑏𝐴𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
4644, 45syl 17 . . . . 5 (𝑔 = (𝑏𝐴𝐺) → ((𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ ((𝑏𝐴𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
4746imbi2d 329 . . . 4 (𝑔 = (𝑏𝐴𝐺) → ((ω ⊆ 𝑏 → (𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏) ↔ (ω ⊆ 𝑏 → ((𝑏𝐴𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)))
4843, 47ralbid 2966 . . 3 (𝑔 = (𝑏𝐴𝐺) → (∀𝑏𝐴 (ω ⊆ 𝑏 → (𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏) ↔ ∀𝑏𝐴 (ω ⊆ 𝑏 → ((𝑏𝐴𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)))
4948spcegv 3267 . 2 ((𝑏𝐴𝐺) ∈ V → (∀𝑏𝐴 (ω ⊆ 𝑏 → ((𝑏𝐴𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto𝑏) → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → (𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)))
503, 41, 49sylc 63 1 (𝜑 → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → (𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wex 1695  wcel 1977  wral 2896  wrex 2897  {crab 2900  Vcvv 3173  cdif 3537  wss 3540  c0 3874  cop 4131   class class class wbr 4583  cmpt 4643   I cid 4948   × cxp 5036  ccnv 5037  ran crn 5039  cres 5040  ccom 5042  Oncon0 5640  wf 5800  1-1-ontowf1o 5803  cfv 5804  (class class class)co 6549  cmpt2 6551  ωcom 6957  1𝑜c1o 7440  2𝑜c2o 7441   +𝑜 coa 7444   ·𝑜 comu 7445  𝑜 coe 7446  𝑚 cmap 7744   finSupp cfsupp 8158   CNF ccnf 8441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-cnf 8442
This theorem is referenced by:  infxpenc2lem3  8727
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