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Theorem cantnflem1b 8543
Description: Lemma for cantnf 8550. (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 795 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑋) ⊆ 𝑢)
2 cantnflem1.o . . . . . . . 8 𝑂 = OrdIso( E , (𝐺 supp ∅))
32oicl 8394 . . . . . . 7 Ord dom 𝑂
4 cantnfs.b . . . . . . . . . . . 12 (𝜑𝐵 ∈ On)
5 suppssdm 7268 . . . . . . . . . . . . 13 (𝐺 supp ∅) ⊆ dom 𝐺
6 oemapval.g . . . . . . . . . . . . . . . 16 (𝜑𝐺𝑆)
7 cantnfs.s . . . . . . . . . . . . . . . . 17 𝑆 = dom (𝐴 CNF 𝐵)
8 cantnfs.a . . . . . . . . . . . . . . . . 17 (𝜑𝐴 ∈ On)
97, 8, 4cantnfs 8523 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
106, 9mpbid 222 . . . . . . . . . . . . . . 15 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
1110simpld 475 . . . . . . . . . . . . . 14 (𝜑𝐺:𝐵𝐴)
12 fdm 6018 . . . . . . . . . . . . . 14 (𝐺:𝐵𝐴 → dom 𝐺 = 𝐵)
1311, 12syl 17 . . . . . . . . . . . . 13 (𝜑 → dom 𝐺 = 𝐵)
145, 13syl5sseq 3638 . . . . . . . . . . . 12 (𝜑 → (𝐺 supp ∅) ⊆ 𝐵)
154, 14ssexd 4775 . . . . . . . . . . 11 (𝜑 → (𝐺 supp ∅) ∈ V)
167, 8, 4, 2, 6cantnfcl 8524 . . . . . . . . . . . 12 (𝜑 → ( E We (𝐺 supp ∅) ∧ dom 𝑂 ∈ ω))
1716simpld 475 . . . . . . . . . . 11 (𝜑 → E We (𝐺 supp ∅))
182oiiso 8402 . . . . . . . . . . 11 (((𝐺 supp ∅) ∈ V ∧ E We (𝐺 supp ∅)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
1915, 17, 18syl2anc 692 . . . . . . . . . 10 (𝜑𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
20 isof1o 6538 . . . . . . . . . 10 (𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)) → 𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅))
2119, 20syl 17 . . . . . . . . 9 (𝜑𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅))
22 f1ocnv 6116 . . . . . . . . 9 (𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) → 𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂)
23 f1of 6104 . . . . . . . . 9 (𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂𝑂:(𝐺 supp ∅)⟶dom 𝑂)
2421, 22, 233syl 18 . . . . . . . 8 (𝜑𝑂:(𝐺 supp ∅)⟶dom 𝑂)
25 oemapval.t . . . . . . . . 9 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
26 oemapval.f . . . . . . . . 9 (𝜑𝐹𝑆)
27 oemapvali.r . . . . . . . . 9 (𝜑𝐹𝑇𝐺)
28 oemapvali.x . . . . . . . . 9 𝑋 = {𝑐𝐵 ∣ (𝐹𝑐) ∈ (𝐺𝑐)}
297, 8, 4, 25, 26, 6, 27, 28cantnflem1a 8542 . . . . . . . 8 (𝜑𝑋 ∈ (𝐺 supp ∅))
3024, 29ffvelrnd 6326 . . . . . . 7 (𝜑 → (𝑂𝑋) ∈ dom 𝑂)
31 ordelon 5716 . . . . . . 7 ((Ord dom 𝑂 ∧ (𝑂𝑋) ∈ dom 𝑂) → (𝑂𝑋) ∈ On)
323, 30, 31sylancr 694 . . . . . 6 (𝜑 → (𝑂𝑋) ∈ On)
3332adantr 481 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑋) ∈ On)
343a1i 11 . . . . . . . 8 (𝜑 → Ord dom 𝑂)
35 ordelon 5716 . . . . . . . 8 ((Ord dom 𝑂 ∧ suc 𝑢 ∈ dom 𝑂) → suc 𝑢 ∈ On)
3634, 35sylan 488 . . . . . . 7 ((𝜑 ∧ suc 𝑢 ∈ dom 𝑂) → suc 𝑢 ∈ On)
37 sucelon 6979 . . . . . . 7 (𝑢 ∈ On ↔ suc 𝑢 ∈ On)
3836, 37sylibr 224 . . . . . 6 ((𝜑 ∧ suc 𝑢 ∈ dom 𝑂) → 𝑢 ∈ On)
3938adantrr 752 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑢 ∈ On)
40 ontri1 5726 . . . . 5 (((𝑂𝑋) ∈ On ∧ 𝑢 ∈ On) → ((𝑂𝑋) ⊆ 𝑢 ↔ ¬ 𝑢 ∈ (𝑂𝑋)))
4133, 39, 40syl2anc 692 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → ((𝑂𝑋) ⊆ 𝑢 ↔ ¬ 𝑢 ∈ (𝑂𝑋)))
421, 41mpbid 222 . . 3 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → ¬ 𝑢 ∈ (𝑂𝑋))
4319adantr 481 . . . . . 6 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
44 ordtr 5706 . . . . . . . 8 (Ord dom 𝑂 → Tr dom 𝑂)
453, 44mp1i 13 . . . . . . 7 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → Tr dom 𝑂)
46 simprl 793 . . . . . . 7 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → suc 𝑢 ∈ dom 𝑂)
47 trsuc 5779 . . . . . . 7 ((Tr dom 𝑂 ∧ suc 𝑢 ∈ dom 𝑂) → 𝑢 ∈ dom 𝑂)
4845, 46, 47syl2anc 692 . . . . . 6 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑢 ∈ dom 𝑂)
4930adantr 481 . . . . . 6 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑋) ∈ dom 𝑂)
50 isorel 6541 . . . . . 6 ((𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)) ∧ (𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ∈ dom 𝑂)) → (𝑢 E (𝑂𝑋) ↔ (𝑂𝑢) E (𝑂‘(𝑂𝑋))))
5143, 48, 49, 50syl12anc 1321 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑢 E (𝑂𝑋) ↔ (𝑂𝑢) E (𝑂‘(𝑂𝑋))))
52 fvex 6168 . . . . . 6 (𝑂𝑋) ∈ V
5352epelc 4997 . . . . 5 (𝑢 E (𝑂𝑋) ↔ 𝑢 ∈ (𝑂𝑋))
54 fvex 6168 . . . . . 6 (𝑂‘(𝑂𝑋)) ∈ V
5554epelc 4997 . . . . 5 ((𝑂𝑢) E (𝑂‘(𝑂𝑋)) ↔ (𝑂𝑢) ∈ (𝑂‘(𝑂𝑋)))
5651, 53, 553bitr3g 302 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑢 ∈ (𝑂𝑋) ↔ (𝑂𝑢) ∈ (𝑂‘(𝑂𝑋))))
57 f1ocnvfv2 6498 . . . . . . 7 ((𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) ∧ 𝑋 ∈ (𝐺 supp ∅)) → (𝑂‘(𝑂𝑋)) = 𝑋)
5821, 29, 57syl2anc 692 . . . . . 6 (𝜑 → (𝑂‘(𝑂𝑋)) = 𝑋)
5958adantr 481 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂‘(𝑂𝑋)) = 𝑋)
6059eleq2d 2684 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → ((𝑂𝑢) ∈ (𝑂‘(𝑂𝑋)) ↔ (𝑂𝑢) ∈ 𝑋))
6156, 60bitrd 268 . . 3 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑢 ∈ (𝑂𝑋) ↔ (𝑂𝑢) ∈ 𝑋))
6242, 61mtbid 314 . 2 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → ¬ (𝑂𝑢) ∈ 𝑋)
637, 8, 4, 25, 26, 6, 27, 28oemapvali 8541 . . . . . 6 (𝜑 → (𝑋𝐵 ∧ (𝐹𝑋) ∈ (𝐺𝑋) ∧ ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤))))
6463simp1d 1071 . . . . 5 (𝜑𝑋𝐵)
65 onelon 5717 . . . . 5 ((𝐵 ∈ On ∧ 𝑋𝐵) → 𝑋 ∈ On)
664, 64, 65syl2anc 692 . . . 4 (𝜑𝑋 ∈ On)
6766adantr 481 . . 3 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑋 ∈ On)
684adantr 481 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝐵 ∈ On)
6914adantr 481 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝐺 supp ∅) ⊆ 𝐵)
702oif 8395 . . . . . . 7 𝑂:dom 𝑂⟶(𝐺 supp ∅)
7170ffvelrni 6324 . . . . . 6 (𝑢 ∈ dom 𝑂 → (𝑂𝑢) ∈ (𝐺 supp ∅))
7248, 71syl 17 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑢) ∈ (𝐺 supp ∅))
7369, 72sseldd 3589 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑢) ∈ 𝐵)
74 onelon 5717 . . . 4 ((𝐵 ∈ On ∧ (𝑂𝑢) ∈ 𝐵) → (𝑂𝑢) ∈ On)
7568, 73, 74syl2anc 692 . . 3 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑢) ∈ On)
76 ontri1 5726 . . 3 ((𝑋 ∈ On ∧ (𝑂𝑢) ∈ On) → (𝑋 ⊆ (𝑂𝑢) ↔ ¬ (𝑂𝑢) ∈ 𝑋))
7767, 75, 76syl2anc 692 . 2 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑋 ⊆ (𝑂𝑢) ↔ ¬ (𝑂𝑢) ∈ 𝑋))
7862, 77mpbird 247 1 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑋 ⊆ (𝑂𝑢))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2908  wrex 2909  {crab 2912  Vcvv 3190  wss 3560  c0 3897   cuni 4409   class class class wbr 4623  {copab 4682  Tr wtr 4722   E cep 4993   We wwe 5042  ccnv 5083  dom cdm 5084  Ord word 5691  Oncon0 5692  suc csuc 5694  wf 5853  1-1-ontowf1o 5856  cfv 5857   Isom wiso 5858  (class class class)co 6615  ωcom 7027   supp csupp 7255   finSupp cfsupp 8235  OrdIsocoi 8374   CNF ccnf 8518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-supp 7256  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-seqom 7503  df-1o 7520  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fsupp 8236  df-oi 8375  df-cnf 8519
This theorem is referenced by:  cantnflem1c  8544
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