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Theorem cnfcom3clem 8485
 Description: Lemma for cnfcom3c 8486. (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 1054 . . . . . 6 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝐴 ∈ On)
3 omelon 8426 . . . . . . . . 9 ω ∈ On
4 1onn 7606 . . . . . . . . 9 1𝑜 ∈ ω
5 ondif2 7469 . . . . . . . . 9 (ω ∈ (On ∖ 2𝑜) ↔ (ω ∈ On ∧ 1𝑜 ∈ ω))
63, 4, 5mpbir2an 957 . . . . . . . 8 ω ∈ (On ∖ 2𝑜)
7 oeworde 7560 . . . . . . . 8 ((ω ∈ (On ∖ 2𝑜) ∧ 𝐴 ∈ On) → 𝐴 ⊆ (ω ↑𝑜 𝐴))
86, 2, 7sylancr 694 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝐴 ⊆ (ω ↑𝑜 𝐴))
9 simp2 1055 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝑏𝐴)
108, 9sseldd 3569 . . . . . 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 1056 . . . . . 6 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → ω ⊆ 𝑏)
191, 2, 10, 11, 12, 13, 14, 15, 16, 17, 18cnfcom3lem 8483 . . . . 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 8484 . . . . . 6 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝑁:𝑏1-1-onto→(ω ↑𝑜 𝑊))
24 f1of 6050 . . . . . . . . . 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 3176 . . . . . . . . 9 𝑏 ∈ V
27 fex 6394 . . . . . . . . 9 ((𝑁:𝑏⟶(ω ↑𝑜 𝑊) ∧ 𝑏 ∈ V) → 𝑁 ∈ V)
2825, 26, 27sylancl 693 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → 𝑁 ∈ V)
29 cnfcom3c.l . . . . . . . . 9 𝐿 = (𝑏 ∈ (ω ↑𝑜 𝐴) ↦ 𝑁)
3029fvmpt2 6200 . . . . . . . 8 ((𝑏 ∈ (ω ↑𝑜 𝐴) ∧ 𝑁 ∈ V) → (𝐿𝑏) = 𝑁)
3110, 28, 30syl2anc 691 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → (𝐿𝑏) = 𝑁)
32 f1oeq1 6040 . . . . . . 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 246 . . . . 5 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → (𝐿𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑊))
35 oveq2 6557 . . . . . . 7 (𝑤 = 𝑊 → (ω ↑𝑜 𝑤) = (ω ↑𝑜 𝑊))
36 f1oeq3 6042 . . . . . . 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 3282 . . . . 5 ((𝑊 ∈ (On ∖ 1𝑜) ∧ (𝐿𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑊)) → ∃𝑤 ∈ (On ∖ 1𝑜)(𝐿𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))
3919, 34, 38syl2anc 691 . . . 4 ((𝐴 ∈ On ∧ 𝑏𝐴 ∧ ω ⊆ 𝑏) → ∃𝑤 ∈ (On ∖ 1𝑜)(𝐿𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤))
40393expia 1259 . . 3 ((𝐴 ∈ On ∧ 𝑏𝐴) → (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝐿𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
4140ralrimiva 2949 . 2 (𝐴 ∈ On → ∀𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1𝑜)(𝐿𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
42 ovex 6577 . . . . 5 (ω ↑𝑜 𝐴) ∈ V
4342mptex 6390 . . . 4 (𝑏 ∈ (ω ↑𝑜 𝐴) ↦ 𝑁) ∈ V
4429, 43eqeltri 2684 . . 3 𝐿 ∈ V
45 nfmpt1 4675 . . . . . 6 𝑏(𝑏 ∈ (ω ↑𝑜 𝐴) ↦ 𝑁)
4629, 45nfcxfr 2749 . . . . 5 𝑏𝐿
4746nfeq2 2766 . . . 4 𝑏 𝑔 = 𝐿
48 fveq1 6102 . . . . . . 7 (𝑔 = 𝐿 → (𝑔𝑏) = (𝐿𝑏))
49 f1oeq1 6040 . . . . . . 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 3034 . . . . 5 (𝑔 = 𝐿 → (∃𝑤 ∈ (On ∖ 1𝑜)(𝑔𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤) ↔ ∃𝑤 ∈ (On ∖ 1𝑜)(𝐿𝑏):𝑏1-1-onto→(ω ↑𝑜 𝑤)))
5251imbi2d 329 . . . 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 2966 . . 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 3273 . 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 195   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372   ↦ cmpt 4643   E cep 4947  ◡ccnv 5037  dom cdm 5038   ∘ ccom 5042  Oncon0 5640  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957   supp csupp 7182  seq𝜔cseqom 7429  1𝑜c1o 7440  2𝑜c2o 7441   +𝑜 coa 7444   ·𝑜 comu 7445   ↑𝑜 coe 7446  OrdIsocoi 8297   CNF ccnf 8441 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-cnf 8442 This theorem is referenced by:  cnfcom3c  8486
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