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Theorem cnfcom3clem 8599
Description: Lemma for cnfcom3c 8600. (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 1060 . . . . . 6 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝐴 ∈ On)
3 omelon 8540 . . . . . . . . 9 ω ∈ On
4 1onn 7716 . . . . . . . . 9 1𝑜 ∈ ω
5 ondif2 7579 . . . . . . . . 9 (ω ∈ (On ∖ 2𝑜) ↔ (ω ∈ On ∧ 1𝑜 ∈ ω))
63, 4, 5mpbir2an 955 . . . . . . . 8 ω ∈ (On ∖ 2𝑜)
7 oeworde 7670 . . . . . . . 8 ((ω ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → 𝐴 ⊆ (ω ↑𝑜 𝐴))
86, 2, 7sylancr 695 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝐴 ⊆ (ω ↑𝑜 𝐴))
9 simp2 1061 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝑏𝐴)
108, 9sseldd 3602 . . . . . 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 1062 . . . . . 6 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → ω ⊆ 𝑏)
191, 2, 10, 11, 12, 13, 14, 15, 16, 17, 18cnfcom3lem 8597 . . . . 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 8598 . . . . . 6 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝑁:𝑏1-1-onto→(ω ↑𝑜 𝑊))
24 f1of 6135 . . . . . . . . . 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 3201 . . . . . . . . 9 𝑏 ∈ V
27 fex 6487 . . . . . . . . 9 ((𝑁:𝑏⟶(ω ↑𝑜 𝑊) ∧ 𝑏 ∈ V) → 𝑁 ∈ V)
2825, 26, 27sylancl 694 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝑁 ∈ V)
29 cnfcom3c.l . . . . . . . . 9 𝐿 = (𝑏 ∈ (ω ↑𝑜 𝐴) ↦ 𝑁)
3029fvmpt2 6289 . . . . . . . 8 ((𝑏 ∈ (ω ↑𝑜 𝐴) ∧ 𝑁 ∈ V) → (𝐿𝑏) = 𝑁)
3110, 28, 30syl2anc 693 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → (𝐿𝑏) = 𝑁)
32 f1oeq1 6125 . . . . . . 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 247 . . . . 5 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → (𝐿𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑊))
35 oveq2 6655 . . . . . . 7 (𝑤 = 𝑊 → (ω ↑𝑜 𝑤) = (ω ↑𝑜 𝑊))
36 f1oeq3 6127 . . . . . . 7 ((ω ↑𝑜 𝑤) = (ω ↑𝑜 𝑊) → ((𝐿𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤) ↔ (𝐿𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑊)))
3735, 36syl 17 . . . . . 6 (𝑤 = 𝑊 → ((𝐿𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤) ↔ (𝐿𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑊)))
3837rspcev 3307 . . . . 5 ((𝑊 ∈ (On ∖ 1𝑜) ∧ (𝐿𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑊)) → ∃𝑤 ∈ (On ∖ 1𝑜)(𝐿𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))
3919, 34, 38syl2anc 693 . . . 4 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → ∃𝑤 ∈ (On ∖ 1𝑜)(𝐿𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))
40393expia 1266 . . 3 ((𝐴 ∈ On ∧ 𝑏𝐴) → (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝐿𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
4140ralrimiva 2965 . 2 (𝐴 ∈ On → ∀𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝐿𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
42 ovex 6675 . . . . 5 (ω ↑𝑜 𝐴) ∈ V
4342mptex 6483 . . . 4 (𝑏 ∈ (ω ↑𝑜 𝐴) ↦ 𝑁) ∈ V
4429, 43eqeltri 2696 . . 3 𝐿 ∈ V
45 nfmpt1 4745 . . . . . 6 𝑏(𝑏 ∈ (ω ↑𝑜 𝐴) ↦ 𝑁)
4629, 45nfcxfr 2761 . . . . 5 𝑏𝐿
4746nfeq2 2779 . . . 4 𝑏 𝑔 = 𝐿
48 fveq1 6188 . . . . . . 7 (𝑔 = 𝐿 → (𝑔𝑏) = (𝐿𝑏))
49 f1oeq1 6125 . . . . . . 7 ((𝑔𝑏) = (𝐿𝑏) → ((𝑔𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤) ↔ (𝐿𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
5048, 49syl 17 . . . . . 6 (𝑔 = 𝐿 → ((𝑔𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤) ↔ (𝐿𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
5150rexbidv 3050 . . . . 5 (𝑔 = 𝐿 → (∃𝑤 ∈ (On ∖ 1𝑜)(𝑔𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤) ↔ ∃𝑤 ∈ (On ∖ 1𝑜)(𝐿𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
5251imbi2d 330 . . . 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))))
5347, 52ralbid 2982 . . 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))))
5444, 53spcev 3298 . 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)))
5541, 54syl 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 196  w3a 1037   = wceq 1482  wex 1703  wcel 1989  wral 2911  wrex 2912  Vcvv 3198  cdif 3569  cun 3570  wss 3572  c0 3913   cuni 4434  cmpt 4727   E cep 5026  ccnv 5111  dom cdm 5112  ccom 5116  Oncon0 5721  wf 5882  1-1-ontowf1o 5885  cfv 5886  (class class class)co 6647  cmpt2 6649  ωcom 7062   supp csupp 7292  seq𝜔cseqom 7539  1𝑜c1o 7550  2𝑜c2o 7551   +𝑜 coa 7554   ·𝑜 comu 7555  𝑜 coe 7556  OrdIsocoi 8411   CNF ccnf 8555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-inf2 8535
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-fal 1488  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-se 5072  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-isom 5895  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-supp 7293  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-seqom 7540  df-1o 7557  df-2o 7558  df-oadd 7561  df-omul 7562  df-oexp 7563  df-er 7739  df-map 7856  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-fsupp 8273  df-oi 8412  df-cnf 8556
This theorem is referenced by:  cnfcom3c  8600
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