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Theorem cnfcom3clem 9742
Description: Lemma for cnfcom3c 9743. (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 1135 . . . . . 6 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝐴 ∈ On)
3 omelon 9683 . . . . . . . . 9 ω ∈ On
4 1onn 8676 . . . . . . . . 9 1o ∈ ω
5 ondif2 8538 . . . . . . . . 9 (ω ∈ (On ∖ 2o) ↔ (ω ∈ On ∧ 1o ∈ ω))
63, 4, 5mpbir2an 711 . . . . . . . 8 ω ∈ (On ∖ 2o)
7 oeworde 8629 . . . . . . . 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 1136 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝑏𝐴)
108, 9sseldd 3995 . . . . . 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 1137 . . . . . 6 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → ω ⊆ 𝑏)
191, 2, 10, 11, 12, 13, 14, 15, 16, 17, 18cnfcom3lem 9740 . . . . 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 9741 . . . . . 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 7246 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝑁 ∈ V)
27 cnfcom3c.l . . . . . . . . 9 𝐿 = (𝑏 ∈ (ω ↑o 𝐴) ↦ 𝑁)
2827fvmpt2 7026 . . . . . . . 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 7438 . . . . . . 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 3621 . . . . 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 1120 . . 3 ((𝐴 ∈ On ∧ 𝑏𝐴) → (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
3736ralrimiva 3143 . 2 (𝐴 ∈ On → ∀𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝐿𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
38 ovex 7463 . . . . 5 (ω ↑o 𝐴) ∈ V
3938mptex 7242 . . . 4 (𝑏 ∈ (ω ↑o 𝐴) ↦ 𝑁) ∈ V
4027, 39eqeltri 2834 . . 3 𝐿 ∈ V
41 nfmpt1 5255 . . . . . 6 𝑏(𝑏 ∈ (ω ↑o 𝐴) ↦ 𝑁)
4227, 41nfcxfr 2900 . . . . 5 𝑏𝐿
4342nfeq2 2920 . . . 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 3176 . . . . 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 3270 . . 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 3605 . 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 1086   = wceq 1536  wex 1775  wcel 2105  wral 3058  wrex 3067  Vcvv 3477  cdif 3959  cun 3960  wss 3962  c0 4338   cuni 4911  cmpt 5230   E cep 5587  ccnv 5687  dom cdm 5688  ccom 5692  Oncon0 6385  wf 6558  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  cmpo 7432  ωcom 7886   supp csupp 8183  seqωcseqom 8485  1oc1o 8497  2oc2o 8498   +o coa 8501   ·o comu 8502  o coe 8503  OrdIsocoi 9546   CNF ccnf 9698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-seqom 8486  df-1o 8504  df-2o 8505  df-oadd 8508  df-omul 8509  df-oexp 8510  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-oi 9547  df-cnf 9699
This theorem is referenced by:  cnfcom3c  9743
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