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Theorem infxpenc2lem2 10061
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‘∅) = ∅)
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)
21mptexd 7245 . 2 (𝜑 → (𝑏𝐴𝐺) ∈ V)
31adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → 𝐴 ∈ On)
4 simprl 770 . . . . . . 7 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → 𝑏𝐴)
5 onelon 6408 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑏𝐴) → 𝑏 ∈ On)
63, 4, 5syl2anc 584 . . . . . 6 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → 𝑏 ∈ On)
7 simprr 772 . . . . . 6 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → ω ⊆ 𝑏)
8 infxpenc2.2 . . . . . . . 8 (𝜑 → ∀𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
9 infxpenc2.3 . . . . . . . 8 𝑊 = ((𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛𝑏))
101, 8, 9infxpenc2lem1 10060 . . . . . . 7 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → (𝑊 ∈ (On ∖ 1o) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑊)))
1110simpld 494 . . . . . 6 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → 𝑊 ∈ (On ∖ 1o))
12 infxpenc2.4 . . . . . . 7 (𝜑𝐹:(ω ↑o 2o)–1-1-onto→ω)
1312adantr 480 . . . . . 6 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → 𝐹:(ω ↑o 2o)–1-1-onto→ω)
14 infxpenc2.5 . . . . . . 7 (𝜑 → (𝐹‘∅) = ∅)
1514adantr 480 . . . . . 6 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → (𝐹‘∅) = ∅)
1610simprd 495 . . . . . 6 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → (𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑊))
17 infxpenc2.k . . . . . 6 𝐾 = (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦( I ↾ 𝑊))))
18 infxpenc2.h . . . . . 6 𝐻 = (((ω CNF 𝑊) ∘ 𝐾) ∘ ((ω ↑o 2o) CNF 𝑊))
19 infxpenc2.l . . . . . 6 𝐿 = (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦(𝑌𝑋))))
20 infxpenc2.x . . . . . 6 𝑋 = (𝑧 ∈ 2o, 𝑤𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))
21 infxpenc2.y . . . . . 6 𝑌 = (𝑧 ∈ 2o, 𝑤𝑊 ↦ ((2o ·o 𝑤) +o 𝑧))
22 infxpenc2.j . . . . . 6 𝐽 = (((ω CNF (2o ·o 𝑊)) ∘ 𝐿) ∘ (ω CNF (𝑊 ·o 2o)))
23 infxpenc2.z . . . . . 6 𝑍 = (𝑥 ∈ (ω ↑o 𝑊), 𝑦 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑥) +o 𝑦))
24 infxpenc2.t . . . . . 6 𝑇 = (𝑥𝑏, 𝑦𝑏 ↦ ⟨((𝑛𝑏)‘𝑥), ((𝑛𝑏)‘𝑦)⟩)
25 infxpenc2.g . . . . . 6 𝐺 = ((𝑛𝑏) ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇))
266, 7, 11, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25infxpenc 10059 . . . . 5 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → 𝐺:(𝑏 × 𝑏)–1-1-onto𝑏)
27 f1of 6847 . . . . . . . . 9 (𝐺:(𝑏 × 𝑏)–1-1-onto𝑏𝐺:(𝑏 × 𝑏)⟶𝑏)
2826, 27syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → 𝐺:(𝑏 × 𝑏)⟶𝑏)
29 vex 3483 . . . . . . . . 9 𝑏 ∈ V
3029, 29xpex 7774 . . . . . . . 8 (𝑏 × 𝑏) ∈ V
31 fex 7247 . . . . . . . 8 ((𝐺:(𝑏 × 𝑏)⟶𝑏 ∧ (𝑏 × 𝑏) ∈ V) → 𝐺 ∈ V)
3228, 30, 31sylancl 586 . . . . . . 7 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → 𝐺 ∈ V)
33 eqid 2736 . . . . . . . 8 (𝑏𝐴𝐺) = (𝑏𝐴𝐺)
3433fvmpt2 7026 . . . . . . 7 ((𝑏𝐴𝐺 ∈ V) → ((𝑏𝐴𝐺)‘𝑏) = 𝐺)
354, 32, 34syl2anc 584 . . . . . 6 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → ((𝑏𝐴𝐺)‘𝑏) = 𝐺)
3635f1oeq1d 6842 . . . . 5 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → (((𝑏𝐴𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto𝑏𝐺:(𝑏 × 𝑏)–1-1-onto𝑏))
3726, 36mpbird 257 . . . 4 ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → ((𝑏𝐴𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)
3837expr 456 . . 3 ((𝜑𝑏𝐴) → (ω ⊆ 𝑏 → ((𝑏𝐴𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
3938ralrimiva 3145 . 2 (𝜑 → ∀𝑏𝐴 (ω ⊆ 𝑏 → ((𝑏𝐴𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
40 nfmpt1 5249 . . . 4 𝑏(𝑏𝐴𝐺)
4140nfeq2 2922 . . 3 𝑏 𝑔 = (𝑏𝐴𝐺)
42 fveq1 6904 . . . . 5 (𝑔 = (𝑏𝐴𝐺) → (𝑔𝑏) = ((𝑏𝐴𝐺)‘𝑏))
4342f1oeq1d 6842 . . . 4 (𝑔 = (𝑏𝐴𝐺) → ((𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏 ↔ ((𝑏𝐴𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
4443imbi2d 340 . . 3 (𝑔 = (𝑏𝐴𝐺) → ((ω ⊆ 𝑏 → (𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏) ↔ (ω ⊆ 𝑏 → ((𝑏𝐴𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)))
4541, 44ralbid 3272 . 2 (𝑔 = (𝑏𝐴𝐺) → (∀𝑏𝐴 (ω ⊆ 𝑏 → (𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏) ↔ ∀𝑏𝐴 (ω ⊆ 𝑏 → ((𝑏𝐴𝐺)‘𝑏):(𝑏 × 𝑏)–1-1-onto𝑏)))
462, 39, 45spcedv 3597 1 (𝜑 → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → (𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wex 1778  wcel 2107  wral 3060  wrex 3069  {crab 3435  Vcvv 3479  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  wf 6556  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:  infxpenc2lem3  10062
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