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Theorem cnfcom3clem 9745
Description: Lemma for cnfcom3c 9746. (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 1137 . . . . . 6 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝐴 ∈ On)
3 omelon 9686 . . . . . . . . 9 ω ∈ On
4 1onn 8678 . . . . . . . . 9 1o ∈ ω
5 ondif2 8540 . . . . . . . . 9 (ω ∈ (On ∖ 2o) ↔ (ω ∈ On ∧ 1o ∈ ω))
63, 4, 5mpbir2an 711 . . . . . . . 8 ω ∈ (On ∖ 2o)
7 oeworde 8631 . . . . . . . 8 ((ω ∈ (On ∖ 2o) ∧ 𝐴 ∈ On) → 𝐴 ⊆ (ω ↑o 𝐴))
86, 2, 7sylancr 587 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝐴 ⊆ (ω ↑o 𝐴))
9 simp2 1138 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝑏𝐴)
108, 9sseldd 3984 . . . . . 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 1139 . . . . . 6 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → ω ⊆ 𝑏)
191, 2, 10, 11, 12, 13, 14, 15, 16, 17, 18cnfcom3lem 9743 . . . . 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 9744 . . . . . 6 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝑁:𝑏1-1-onto→(ω ↑o 𝑊))
24 f1of 6848 . . . . . . . . . 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))
2625, 9fexd 7247 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝑁 ∈ V)
27 cnfcom3c.l . . . . . . . . 9 𝐿 = (𝑏 ∈ (ω ↑o 𝐴) ↦ 𝑁)
2827fvmpt2 7027 . . . . . . . 8 ((𝑏 ∈ (ω ↑o 𝐴) ∧ 𝑁 ∈ V) → (𝐿𝑏) = 𝑁)
2910, 26, 28syl2anc 584 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → (𝐿𝑏) = 𝑁)
3029f1oeq1d 6843 . . . . . 6 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → ((𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑊) ↔ 𝑁:𝑏1-1-onto→(ω ↑o 𝑊)))
3123, 30mpbird 257 . . . . 5 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → (𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑊))
32 oveq2 7439 . . . . . . 7 (𝑤 = 𝑊 → (ω ↑o 𝑤) = (ω ↑o 𝑊))
3332f1oeq3d 6845 . . . . . 6 (𝑤 = 𝑊 → ((𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑤) ↔ (𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑊)))
3433rspcev 3622 . . . . 5 ((𝑊 ∈ (On ∖ 1o) ∧ (𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑊)) → ∃𝑤 ∈ (On ∖ 1o)(𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑤))
3519, 31, 34syl2anc 584 . . . 4 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → ∃𝑤 ∈ (On ∖ 1o)(𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑤))
36353expia 1122 . . 3 ((𝐴 ∈ On ∧ 𝑏𝐴) → (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
3736ralrimiva 3146 . 2 (𝐴 ∈ On → ∀𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
38 ovex 7464 . . . . 5 (ω ↑o 𝐴) ∈ V
3938mptex 7243 . . . 4 (𝑏 ∈ (ω ↑o 𝐴) ↦ 𝑁) ∈ V
4027, 39eqeltri 2837 . . 3 𝐿 ∈ V
41 nfmpt1 5250 . . . . . 6 𝑏(𝑏 ∈ (ω ↑o 𝐴) ↦ 𝑁)
4227, 41nfcxfr 2903 . . . . 5 𝑏𝐿
4342nfeq2 2923 . . . 4 𝑏 𝑔 = 𝐿
44 fveq1 6905 . . . . . . 7 (𝑔 = 𝐿 → (𝑔𝑏) = (𝐿𝑏))
4544f1oeq1d 6843 . . . . . 6 (𝑔 = 𝐿 → ((𝑔𝑏):𝑏1-1-onto→(ω ↑o 𝑤) ↔ (𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
4645rexbidv 3179 . . . . 5 (𝑔 = 𝐿 → (∃𝑤 ∈ (On ∖ 1o)(𝑔𝑏):𝑏1-1-onto→(ω ↑o 𝑤) ↔ ∃𝑤 ∈ (On ∖ 1o)(𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
4746imbi2d 340 . . . 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))))
4843, 47ralbid 3273 . . 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))))
4940, 48spcev 3606 . 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)))
5037, 49syl 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  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  cdif 3948  cun 3949  wss 3951  c0 4333   cuni 4907  cmpt 5225   E cep 5583  ccnv 5684  dom cdm 5685  ccom 5689  Oncon0 6384  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  cmpo 7433  ωcom 7887   supp csupp 8185  seqωcseqom 8487  1oc1o 8499  2oc2o 8500   +o coa 8503   ·o comu 8504  o coe 8505  OrdIsocoi 9549   CNF ccnf 9701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seqom 8488  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-oexp 8512  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-oi 9550  df-cnf 9702
This theorem is referenced by:  cnfcom3c  9746
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