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Theorem cantnflem1b 9301
Description: Lemma for cantnf 9308. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)
Hypotheses
Ref Expression
cantnfs.s 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a (𝜑𝐴 ∈ On)
cantnfs.b (𝜑𝐵 ∈ On)
oemapval.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
oemapval.f (𝜑𝐹𝑆)
oemapval.g (𝜑𝐺𝑆)
oemapvali.r (𝜑𝐹𝑇𝐺)
oemapvali.x 𝑋 = {𝑐𝐵 ∣ (𝐹𝑐) ∈ (𝐺𝑐)}
cantnflem1.o 𝑂 = OrdIso( E , (𝐺 supp ∅))
Assertion
Ref Expression
cantnflem1b ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑋 ⊆ (𝑂𝑢))
Distinct variable groups:   𝑢,𝑐,𝑤,𝑥,𝑦,𝑧,𝐵   𝐴,𝑐,𝑢,𝑤,𝑥,𝑦,𝑧   𝑇,𝑐,𝑢   𝑢,𝐹,𝑤,𝑥,𝑦,𝑧   𝑆,𝑐,𝑢,𝑥,𝑦,𝑧   𝐺,𝑐,𝑢,𝑤,𝑥,𝑦,𝑧   𝑢,𝑂,𝑤,𝑥,𝑦,𝑧   𝜑,𝑢,𝑥,𝑦,𝑧   𝑢,𝑋,𝑤,𝑥,𝑦,𝑧   𝐹,𝑐   𝜑,𝑐
Allowed substitution hints:   𝜑(𝑤)   𝑆(𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)   𝑂(𝑐)   𝑋(𝑐)

Proof of Theorem cantnflem1b
StepHypRef Expression
1 simprr 773 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑋) ⊆ 𝑢)
2 cantnflem1.o . . . . . . 7 𝑂 = OrdIso( E , (𝐺 supp ∅))
32oicl 9145 . . . . . 6 Ord dom 𝑂
4 ovexd 7248 . . . . . . . . . 10 (𝜑 → (𝐺 supp ∅) ∈ V)
5 cantnfs.s . . . . . . . . . . . 12 𝑆 = dom (𝐴 CNF 𝐵)
6 cantnfs.a . . . . . . . . . . . 12 (𝜑𝐴 ∈ On)
7 cantnfs.b . . . . . . . . . . . 12 (𝜑𝐵 ∈ On)
8 oemapval.g . . . . . . . . . . . 12 (𝜑𝐺𝑆)
95, 6, 7, 2, 8cantnfcl 9282 . . . . . . . . . . 11 (𝜑 → ( E We (𝐺 supp ∅) ∧ dom 𝑂 ∈ ω))
109simpld 498 . . . . . . . . . 10 (𝜑 → E We (𝐺 supp ∅))
112oiiso 9153 . . . . . . . . . 10 (((𝐺 supp ∅) ∈ V ∧ E We (𝐺 supp ∅)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
124, 10, 11syl2anc 587 . . . . . . . . 9 (𝜑𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
13 isof1o 7132 . . . . . . . . 9 (𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)) → 𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅))
1412, 13syl 17 . . . . . . . 8 (𝜑𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅))
15 f1ocnv 6673 . . . . . . . 8 (𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) → 𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂)
16 f1of 6661 . . . . . . . 8 (𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂𝑂:(𝐺 supp ∅)⟶dom 𝑂)
1714, 15, 163syl 18 . . . . . . 7 (𝜑𝑂:(𝐺 supp ∅)⟶dom 𝑂)
18 oemapval.t . . . . . . . 8 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
19 oemapval.f . . . . . . . 8 (𝜑𝐹𝑆)
20 oemapvali.r . . . . . . . 8 (𝜑𝐹𝑇𝐺)
21 oemapvali.x . . . . . . . 8 𝑋 = {𝑐𝐵 ∣ (𝐹𝑐) ∈ (𝐺𝑐)}
225, 6, 7, 18, 19, 8, 20, 21cantnflem1a 9300 . . . . . . 7 (𝜑𝑋 ∈ (𝐺 supp ∅))
2317, 22ffvelrnd 6905 . . . . . 6 (𝜑 → (𝑂𝑋) ∈ dom 𝑂)
24 ordelon 6237 . . . . . 6 ((Ord dom 𝑂 ∧ (𝑂𝑋) ∈ dom 𝑂) → (𝑂𝑋) ∈ On)
253, 23, 24sylancr 590 . . . . 5 (𝜑 → (𝑂𝑋) ∈ On)
263a1i 11 . . . . . . . 8 (𝜑 → Ord dom 𝑂)
27 ordelon 6237 . . . . . . . 8 ((Ord dom 𝑂 ∧ suc 𝑢 ∈ dom 𝑂) → suc 𝑢 ∈ On)
2826, 27sylan 583 . . . . . . 7 ((𝜑 ∧ suc 𝑢 ∈ dom 𝑂) → suc 𝑢 ∈ On)
29 sucelon 7596 . . . . . . 7 (𝑢 ∈ On ↔ suc 𝑢 ∈ On)
3028, 29sylibr 237 . . . . . 6 ((𝜑 ∧ suc 𝑢 ∈ dom 𝑂) → 𝑢 ∈ On)
3130adantrr 717 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑢 ∈ On)
32 ontri1 6247 . . . . 5 (((𝑂𝑋) ∈ On ∧ 𝑢 ∈ On) → ((𝑂𝑋) ⊆ 𝑢 ↔ ¬ 𝑢 ∈ (𝑂𝑋)))
3325, 31, 32syl2an2r 685 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → ((𝑂𝑋) ⊆ 𝑢 ↔ ¬ 𝑢 ∈ (𝑂𝑋)))
341, 33mpbid 235 . . 3 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → ¬ 𝑢 ∈ (𝑂𝑋))
3512adantr 484 . . . . . 6 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
36 ordtr 6227 . . . . . . . 8 (Ord dom 𝑂 → Tr dom 𝑂)
373, 36mp1i 13 . . . . . . 7 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → Tr dom 𝑂)
38 simprl 771 . . . . . . 7 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → suc 𝑢 ∈ dom 𝑂)
39 trsuc 6297 . . . . . . 7 ((Tr dom 𝑂 ∧ suc 𝑢 ∈ dom 𝑂) → 𝑢 ∈ dom 𝑂)
4037, 38, 39syl2anc 587 . . . . . 6 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑢 ∈ dom 𝑂)
4123adantr 484 . . . . . 6 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑋) ∈ dom 𝑂)
42 isorel 7135 . . . . . 6 ((𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)) ∧ (𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ∈ dom 𝑂)) → (𝑢 E (𝑂𝑋) ↔ (𝑂𝑢) E (𝑂‘(𝑂𝑋))))
4335, 40, 41, 42syl12anc 837 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑢 E (𝑂𝑋) ↔ (𝑂𝑢) E (𝑂‘(𝑂𝑋))))
44 fvex 6730 . . . . . 6 (𝑂𝑋) ∈ V
4544epeli 5462 . . . . 5 (𝑢 E (𝑂𝑋) ↔ 𝑢 ∈ (𝑂𝑋))
46 fvex 6730 . . . . . 6 (𝑂‘(𝑂𝑋)) ∈ V
4746epeli 5462 . . . . 5 ((𝑂𝑢) E (𝑂‘(𝑂𝑋)) ↔ (𝑂𝑢) ∈ (𝑂‘(𝑂𝑋)))
4843, 45, 473bitr3g 316 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑢 ∈ (𝑂𝑋) ↔ (𝑂𝑢) ∈ (𝑂‘(𝑂𝑋))))
49 f1ocnvfv2 7088 . . . . . . 7 ((𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) ∧ 𝑋 ∈ (𝐺 supp ∅)) → (𝑂‘(𝑂𝑋)) = 𝑋)
5014, 22, 49syl2anc 587 . . . . . 6 (𝜑 → (𝑂‘(𝑂𝑋)) = 𝑋)
5150adantr 484 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂‘(𝑂𝑋)) = 𝑋)
5251eleq2d 2823 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → ((𝑂𝑢) ∈ (𝑂‘(𝑂𝑋)) ↔ (𝑂𝑢) ∈ 𝑋))
5348, 52bitrd 282 . . 3 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑢 ∈ (𝑂𝑋) ↔ (𝑂𝑢) ∈ 𝑋))
5434, 53mtbid 327 . 2 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → ¬ (𝑂𝑢) ∈ 𝑋)
555, 6, 7, 18, 19, 8, 20, 21oemapvali 9299 . . . . 5 (𝜑 → (𝑋𝐵 ∧ (𝐹𝑋) ∈ (𝐺𝑋) ∧ ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤))))
5655simp1d 1144 . . . 4 (𝜑𝑋𝐵)
57 onelon 6238 . . . 4 ((𝐵 ∈ On ∧ 𝑋𝐵) → 𝑋 ∈ On)
587, 56, 57syl2anc 587 . . 3 (𝜑𝑋 ∈ On)
59 suppssdm 7919 . . . . . . 7 (𝐺 supp ∅) ⊆ dom 𝐺
605, 6, 7cantnfs 9281 . . . . . . . . 9 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
618, 60mpbid 235 . . . . . . . 8 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
6261simpld 498 . . . . . . 7 (𝜑𝐺:𝐵𝐴)
6359, 62fssdm 6565 . . . . . 6 (𝜑 → (𝐺 supp ∅) ⊆ 𝐵)
6463adantr 484 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝐺 supp ∅) ⊆ 𝐵)
652oif 9146 . . . . . . 7 𝑂:dom 𝑂⟶(𝐺 supp ∅)
6665ffvelrni 6903 . . . . . 6 (𝑢 ∈ dom 𝑂 → (𝑂𝑢) ∈ (𝐺 supp ∅))
6740, 66syl 17 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑢) ∈ (𝐺 supp ∅))
6864, 67sseldd 3902 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑢) ∈ 𝐵)
69 onelon 6238 . . . 4 ((𝐵 ∈ On ∧ (𝑂𝑢) ∈ 𝐵) → (𝑂𝑢) ∈ On)
707, 68, 69syl2an2r 685 . . 3 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑢) ∈ On)
71 ontri1 6247 . . 3 ((𝑋 ∈ On ∧ (𝑂𝑢) ∈ On) → (𝑋 ⊆ (𝑂𝑢) ↔ ¬ (𝑂𝑢) ∈ 𝑋))
7258, 70, 71syl2an2r 685 . 2 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑋 ⊆ (𝑂𝑢) ↔ ¬ (𝑂𝑢) ∈ 𝑋))
7354, 72mpbird 260 1 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑋 ⊆ (𝑂𝑢))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wral 3061  wrex 3062  {crab 3065  Vcvv 3408  wss 3866  c0 4237   cuni 4819   class class class wbr 5053  {copab 5115  Tr wtr 5161   E cep 5459   We wwe 5508  ccnv 5550  dom cdm 5551  Ord word 6212  Oncon0 6213  suc csuc 6215  wf 6376  1-1-ontowf1o 6379  cfv 6380   Isom wiso 6381  (class class class)co 7213  ωcom 7644   supp csupp 7903   finSupp cfsupp 8985  OrdIsocoi 9125   CNF ccnf 9276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-supp 7904  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-seqom 8184  df-1o 8202  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fsupp 8986  df-oi 9126  df-cnf 9277
This theorem is referenced by:  cantnflem1c  9302
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