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Theorem cnfcom3clem 8962
Description: Lemma for cnfcom3c 8963. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 4-Jul-2019.)
Hypotheses
Ref Expression
cnfcom3c.s 𝑆 = dom (ω CNF 𝐴)
cnfcom3c.f 𝐹 = ((ω CNF 𝐴)‘𝑏)
cnfcom3c.g 𝐺 = OrdIso( E , (𝐹 supp ∅))
cnfcom3c.h 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)
cnfcom3c.t 𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
cnfcom3c.m 𝑀 = ((ω ↑o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘)))
cnfcom3c.k 𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))
cnfcom3c.w 𝑊 = (𝐺 dom 𝐺)
cnfcom3c.x 𝑋 = (𝑢 ∈ (𝐹𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((𝐹𝑊) ·o 𝑣) +o 𝑢))
cnfcom3c.y 𝑌 = (𝑢 ∈ (𝐹𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑢) +o 𝑣))
cnfcom3c.n 𝑁 = ((𝑋𝑌) ∘ (𝑇‘dom 𝐺))
cnfcom3c.l 𝐿 = (𝑏 ∈ (ω ↑o 𝐴) ↦ 𝑁)
Assertion
Ref Expression
cnfcom3clem (𝐴 ∈ On → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑔𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
Distinct variable groups:   𝑔,𝑏,𝑘,𝑢,𝑣,𝑤,𝑥,𝑧,𝐴   𝑢,𝐾,𝑣   𝑔,𝐿,𝑤   𝑥,𝑀   𝑢,𝑇,𝑣,𝑧   𝑓,𝑘,𝑢,𝑣,𝑥,𝑧,𝐹   𝑓,𝐺,𝑘,𝑢,𝑣,𝑥,𝑧   𝑓,𝐻,𝑢,𝑣,𝑥   𝑆,𝑘,𝑧   𝑢,𝑊,𝑣,𝑤,𝑥
Allowed substitution hints:   𝐴(𝑓)   𝑆(𝑥,𝑤,𝑣,𝑢,𝑓,𝑔,𝑏)   𝑇(𝑥,𝑤,𝑓,𝑔,𝑘,𝑏)   𝐹(𝑤,𝑔,𝑏)   𝐺(𝑤,𝑔,𝑏)   𝐻(𝑧,𝑤,𝑔,𝑘,𝑏)   𝐾(𝑥,𝑧,𝑤,𝑓,𝑔,𝑘,𝑏)   𝐿(𝑥,𝑧,𝑣,𝑢,𝑓,𝑘,𝑏)   𝑀(𝑧,𝑤,𝑣,𝑢,𝑓,𝑔,𝑘,𝑏)   𝑁(𝑥,𝑧,𝑤,𝑣,𝑢,𝑓,𝑔,𝑘,𝑏)   𝑊(𝑧,𝑓,𝑔,𝑘,𝑏)   𝑋(𝑥,𝑧,𝑤,𝑣,𝑢,𝑓,𝑔,𝑘,𝑏)   𝑌(𝑥,𝑧,𝑤,𝑣,𝑢,𝑓,𝑔,𝑘,𝑏)

Proof of Theorem cnfcom3clem
StepHypRef Expression
1 cnfcom3c.s . . . . . 6 𝑆 = dom (ω CNF 𝐴)
2 simp1 1116 . . . . . 6 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝐴 ∈ On)
3 omelon 8903 . . . . . . . . 9 ω ∈ On
4 1onn 8066 . . . . . . . . 9 1o ∈ ω
5 ondif2 7929 . . . . . . . . 9 (ω ∈ (On ∖ 2o) ↔ (ω ∈ On ∧ 1o ∈ ω))
63, 4, 5mpbir2an 698 . . . . . . . 8 ω ∈ (On ∖ 2o)
7 oeworde 8020 . . . . . . . 8 ((ω ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → 𝐴 ⊆ (ω ↑o 𝐴))
86, 2, 7sylancr 578 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝐴 ⊆ (ω ↑o 𝐴))
9 simp2 1117 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝑏𝐴)
108, 9sseldd 3859 . . . . . 6 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝑏 ∈ (ω ↑o 𝐴))
11 cnfcom3c.f . . . . . 6 𝐹 = ((ω CNF 𝐴)‘𝑏)
12 cnfcom3c.g . . . . . 6 𝐺 = OrdIso( E , (𝐹 supp ∅))
13 cnfcom3c.h . . . . . 6 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)
14 cnfcom3c.t . . . . . 6 𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
15 cnfcom3c.m . . . . . 6 𝑀 = ((ω ↑o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘)))
16 cnfcom3c.k . . . . . 6 𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))
17 cnfcom3c.w . . . . . 6 𝑊 = (𝐺 dom 𝐺)
18 simp3 1118 . . . . . 6 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → ω ⊆ 𝑏)
191, 2, 10, 11, 12, 13, 14, 15, 16, 17, 18cnfcom3lem 8960 . . . . 5 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝑊 ∈ (On ∖ 1o))
20 cnfcom3c.x . . . . . . 7 𝑋 = (𝑢 ∈ (𝐹𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((𝐹𝑊) ·o 𝑣) +o 𝑢))
21 cnfcom3c.y . . . . . . 7 𝑌 = (𝑢 ∈ (𝐹𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑢) +o 𝑣))
22 cnfcom3c.n . . . . . . 7 𝑁 = ((𝑋𝑌) ∘ (𝑇‘dom 𝐺))
231, 2, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22cnfcom3 8961 . . . . . 6 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝑁:𝑏1-1-onto→(ω ↑o 𝑊))
24 f1of 6444 . . . . . . . . . 10 (𝑁:𝑏1-1-onto→(ω ↑o 𝑊) → 𝑁:𝑏⟶(ω ↑o 𝑊))
2523, 24syl 17 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝑁:𝑏⟶(ω ↑o 𝑊))
26 vex 3418 . . . . . . . . 9 𝑏 ∈ V
27 fex 6815 . . . . . . . . 9 ((𝑁:𝑏⟶(ω ↑o 𝑊) ∧ 𝑏 ∈ V) → 𝑁 ∈ V)
2825, 26, 27sylancl 577 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝑁 ∈ V)
29 cnfcom3c.l . . . . . . . . 9 𝐿 = (𝑏 ∈ (ω ↑o 𝐴) ↦ 𝑁)
3029fvmpt2 6605 . . . . . . . 8 ((𝑏 ∈ (ω ↑o 𝐴) ∧ 𝑁 ∈ V) → (𝐿𝑏) = 𝑁)
3110, 28, 30syl2anc 576 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → (𝐿𝑏) = 𝑁)
32 f1oeq1 6433 . . . . . . 7 ((𝐿𝑏) = 𝑁 → ((𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑊) ↔ 𝑁:𝑏1-1-onto→(ω ↑o 𝑊)))
3331, 32syl 17 . . . . . 6 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → ((𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑊) ↔ 𝑁:𝑏1-1-onto→(ω ↑o 𝑊)))
3423, 33mpbird 249 . . . . 5 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → (𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑊))
35 oveq2 6984 . . . . . . 7 (𝑤 = 𝑊 → (ω ↑o 𝑤) = (ω ↑o 𝑊))
3635f1oeq3d 6441 . . . . . 6 (𝑤 = 𝑊 → ((𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑤) ↔ (𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑊)))
3736rspcev 3535 . . . . 5 ((𝑊 ∈ (On ∖ 1o) ∧ (𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑊)) → ∃𝑤 ∈ (On ∖ 1o)(𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑤))
3819, 34, 37syl2anc 576 . . . 4 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → ∃𝑤 ∈ (On ∖ 1o)(𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑤))
39383expia 1101 . . 3 ((𝐴 ∈ On ∧ 𝑏𝐴) → (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
4039ralrimiva 3132 . 2 (𝐴 ∈ On → ∀𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
41 ovex 7008 . . . . 5 (ω ↑o 𝐴) ∈ V
4241mptex 6812 . . . 4 (𝑏 ∈ (ω ↑o 𝐴) ↦ 𝑁) ∈ V
4329, 42eqeltri 2862 . . 3 𝐿 ∈ V
44 nfmpt1 5025 . . . . . 6 𝑏(𝑏 ∈ (ω ↑o 𝐴) ↦ 𝑁)
4529, 44nfcxfr 2930 . . . . 5 𝑏𝐿
4645nfeq2 2947 . . . 4 𝑏 𝑔 = 𝐿
47 fveq1 6498 . . . . . . 7 (𝑔 = 𝐿 → (𝑔𝑏) = (𝐿𝑏))
48 f1oeq1 6433 . . . . . . 7 ((𝑔𝑏) = (𝐿𝑏) → ((𝑔𝑏):𝑏1-1-onto→(ω ↑o 𝑤) ↔ (𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
4947, 48syl 17 . . . . . 6 (𝑔 = 𝐿 → ((𝑔𝑏):𝑏1-1-onto→(ω ↑o 𝑤) ↔ (𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
5049rexbidv 3242 . . . . 5 (𝑔 = 𝐿 → (∃𝑤 ∈ (On ∖ 1o)(𝑔𝑏):𝑏1-1-onto→(ω ↑o 𝑤) ↔ ∃𝑤 ∈ (On ∖ 1o)(𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
5150imbi2d 333 . . . 4 (𝑔 = 𝐿 → ((ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑔𝑏):𝑏1-1-onto→(ω ↑o 𝑤)) ↔ (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑤))))
5246, 51ralbid 3178 . . 3 (𝑔 = 𝐿 → (∀𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑔𝑏):𝑏1-1-onto→(ω ↑o 𝑤)) ↔ ∀𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑤))))
5343, 52spcev 3525 . 2 (∀𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑤)) → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑔𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
5440, 53syl 17 1 (𝐴 ∈ On → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑔𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1068   = wceq 1507  wex 1742  wcel 2050  wral 3088  wrex 3089  Vcvv 3415  cdif 3826  cun 3827  wss 3829  c0 4178   cuni 4712  cmpt 5008   E cep 5316  ccnv 5406  dom cdm 5407  ccom 5411  Oncon0 6029  wf 6184  1-1-ontowf1o 6187  cfv 6188  (class class class)co 6976  cmpo 6978  ωcom 7396   supp csupp 7633  seq𝜔cseqom 7886  1oc1o 7898  2oc2o 7899   +o coa 7902   ·o comu 7903  o coe 7904  OrdIsocoi 8768   CNF ccnf 8918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-inf2 8898
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-se 5367  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-isom 6197  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-supp 7634  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-seqom 7887  df-1o 7905  df-2o 7906  df-oadd 7909  df-omul 7910  df-oexp 7911  df-er 8089  df-map 8208  df-en 8307  df-dom 8308  df-sdom 8309  df-fin 8310  df-fsupp 8629  df-oi 8769  df-cnf 8919
This theorem is referenced by:  cnfcom3c  8963
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