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Theorem cantnflem1b 8696
Description: Lemma for cantnf 8703. (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 813 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑋) ⊆ 𝑢)
2 cantnflem1.o . . . . . . . 8 𝑂 = OrdIso( E , (𝐺 supp ∅))
32oicl 8550 . . . . . . 7 Ord dom 𝑂
4 cantnfs.b . . . . . . . . . . . 12 (𝜑𝐵 ∈ On)
5 suppssdm 7428 . . . . . . . . . . . . 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 8676 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
106, 9mpbid 222 . . . . . . . . . . . . . . 15 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
1110simpld 477 . . . . . . . . . . . . . 14 (𝜑𝐺:𝐵𝐴)
12 fdm 6164 . . . . . . . . . . . . . 14 (𝐺:𝐵𝐴 → dom 𝐺 = 𝐵)
1311, 12syl 17 . . . . . . . . . . . . 13 (𝜑 → dom 𝐺 = 𝐵)
145, 13syl5sseq 3759 . . . . . . . . . . . 12 (𝜑 → (𝐺 supp ∅) ⊆ 𝐵)
154, 14ssexd 4913 . . . . . . . . . . 11 (𝜑 → (𝐺 supp ∅) ∈ V)
167, 8, 4, 2, 6cantnfcl 8677 . . . . . . . . . . . 12 (𝜑 → ( E We (𝐺 supp ∅) ∧ dom 𝑂 ∈ ω))
1716simpld 477 . . . . . . . . . . 11 (𝜑 → E We (𝐺 supp ∅))
182oiiso 8558 . . . . . . . . . . 11 (((𝐺 supp ∅) ∈ V ∧ E We (𝐺 supp ∅)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
1915, 17, 18syl2anc 696 . . . . . . . . . 10 (𝜑𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
20 isof1o 6688 . . . . . . . . . 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 6262 . . . . . . . . 9 (𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) → 𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂)
23 f1of 6250 . . . . . . . . 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 8695 . . . . . . . 8 (𝜑𝑋 ∈ (𝐺 supp ∅))
3024, 29ffvelrnd 6475 . . . . . . 7 (𝜑 → (𝑂𝑋) ∈ dom 𝑂)
31 ordelon 5860 . . . . . . 7 ((Ord dom 𝑂 ∧ (𝑂𝑋) ∈ dom 𝑂) → (𝑂𝑋) ∈ On)
323, 30, 31sylancr 698 . . . . . 6 (𝜑 → (𝑂𝑋) ∈ On)
3332adantr 472 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑋) ∈ On)
343a1i 11 . . . . . . . 8 (𝜑 → Ord dom 𝑂)
35 ordelon 5860 . . . . . . . 8 ((Ord dom 𝑂 ∧ suc 𝑢 ∈ dom 𝑂) → suc 𝑢 ∈ On)
3634, 35sylan 489 . . . . . . 7 ((𝜑 ∧ suc 𝑢 ∈ dom 𝑂) → suc 𝑢 ∈ On)
37 sucelon 7134 . . . . . . 7 (𝑢 ∈ On ↔ suc 𝑢 ∈ On)
3836, 37sylibr 224 . . . . . 6 ((𝜑 ∧ suc 𝑢 ∈ dom 𝑂) → 𝑢 ∈ On)
3938adantrr 755 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑢 ∈ On)
40 ontri1 5870 . . . . 5 (((𝑂𝑋) ∈ On ∧ 𝑢 ∈ On) → ((𝑂𝑋) ⊆ 𝑢 ↔ ¬ 𝑢 ∈ (𝑂𝑋)))
4133, 39, 40syl2anc 696 . . . 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 472 . . . . . 6 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
44 ordtr 5850 . . . . . . . 8 (Ord dom 𝑂 → Tr dom 𝑂)
453, 44mp1i 13 . . . . . . 7 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → Tr dom 𝑂)
46 simprl 811 . . . . . . 7 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → suc 𝑢 ∈ dom 𝑂)
47 trsuc 5923 . . . . . . 7 ((Tr dom 𝑂 ∧ suc 𝑢 ∈ dom 𝑂) → 𝑢 ∈ dom 𝑂)
4845, 46, 47syl2anc 696 . . . . . 6 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑢 ∈ dom 𝑂)
4930adantr 472 . . . . . 6 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑋) ∈ dom 𝑂)
50 isorel 6691 . . . . . 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 1437 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑢 E (𝑂𝑋) ↔ (𝑂𝑢) E (𝑂‘(𝑂𝑋))))
52 fvex 6314 . . . . . 6 (𝑂𝑋) ∈ V
5352epelc 5135 . . . . 5 (𝑢 E (𝑂𝑋) ↔ 𝑢 ∈ (𝑂𝑋))
54 fvex 6314 . . . . . 6 (𝑂‘(𝑂𝑋)) ∈ V
5554epelc 5135 . . . . 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 6648 . . . . . . 7 ((𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) ∧ 𝑋 ∈ (𝐺 supp ∅)) → (𝑂‘(𝑂𝑋)) = 𝑋)
5821, 29, 57syl2anc 696 . . . . . 6 (𝜑 → (𝑂‘(𝑂𝑋)) = 𝑋)
5958adantr 472 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂‘(𝑂𝑋)) = 𝑋)
6059eleq2d 2789 . . . 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 313 . 2 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → ¬ (𝑂𝑢) ∈ 𝑋)
637, 8, 4, 25, 26, 6, 27, 28oemapvali 8694 . . . . . 6 (𝜑 → (𝑋𝐵 ∧ (𝐹𝑋) ∈ (𝐺𝑋) ∧ ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤))))
6463simp1d 1134 . . . . 5 (𝜑𝑋𝐵)
65 onelon 5861 . . . . 5 ((𝐵 ∈ On ∧ 𝑋𝐵) → 𝑋 ∈ On)
664, 64, 65syl2anc 696 . . . 4 (𝜑𝑋 ∈ On)
6766adantr 472 . . 3 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑋 ∈ On)
684adantr 472 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝐵 ∈ On)
6914adantr 472 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝐺 supp ∅) ⊆ 𝐵)
702oif 8551 . . . . . . 7 𝑂:dom 𝑂⟶(𝐺 supp ∅)
7170ffvelrni 6473 . . . . . 6 (𝑢 ∈ dom 𝑂 → (𝑂𝑢) ∈ (𝐺 supp ∅))
7248, 71syl 17 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑢) ∈ (𝐺 supp ∅))
7369, 72sseldd 3710 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑢) ∈ 𝐵)
74 onelon 5861 . . . 4 ((𝐵 ∈ On ∧ (𝑂𝑢) ∈ 𝐵) → (𝑂𝑢) ∈ On)
7568, 73, 74syl2anc 696 . . 3 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑢) ∈ On)
76 ontri1 5870 . . 3 ((𝑋 ∈ On ∧ (𝑂𝑢) ∈ On) → (𝑋 ⊆ (𝑂𝑢) ↔ ¬ (𝑂𝑢) ∈ 𝑋))
7767, 75, 76syl2anc 696 . 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 383   = wceq 1596  wcel 2103  wral 3014  wrex 3015  {crab 3018  Vcvv 3304  wss 3680  c0 4023   cuni 4544   class class class wbr 4760  {copab 4820  Tr wtr 4860   E cep 5132   We wwe 5176  ccnv 5217  dom cdm 5218  Ord word 5835  Oncon0 5836  suc csuc 5838  wf 5997  1-1-ontowf1o 6000  cfv 6001   Isom wiso 6002  (class class class)co 6765  ωcom 7182   supp csupp 7415   finSupp cfsupp 8391  OrdIsocoi 8530   CNF ccnf 8671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-se 5178  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-supp 7416  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-seqom 7663  df-1o 7680  df-er 7862  df-map 7976  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-fsupp 8392  df-oi 8531  df-cnf 8672
This theorem is referenced by:  cantnflem1c  8697
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