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Theorem cantnflem1b 8995
Description: Lemma for cantnf 9002. (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 769 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑋) ⊆ 𝑢)
2 cantnflem1.o . . . . . . 7 𝑂 = OrdIso( E , (𝐺 supp ∅))
32oicl 8839 . . . . . 6 Ord dom 𝑂
4 ovexd 7050 . . . . . . . . . 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 8976 . . . . . . . . . . 11 (𝜑 → ( E We (𝐺 supp ∅) ∧ dom 𝑂 ∈ ω))
109simpld 495 . . . . . . . . . 10 (𝜑 → E We (𝐺 supp ∅))
112oiiso 8847 . . . . . . . . . 10 (((𝐺 supp ∅) ∈ V ∧ E We (𝐺 supp ∅)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
124, 10, 11syl2anc 584 . . . . . . . . 9 (𝜑𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
13 isof1o 6939 . . . . . . . . 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 6495 . . . . . . . 8 (𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) → 𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂)
16 f1of 6483 . . . . . . . 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 8994 . . . . . . 7 (𝜑𝑋 ∈ (𝐺 supp ∅))
2317, 22ffvelrnd 6717 . . . . . 6 (𝜑 → (𝑂𝑋) ∈ dom 𝑂)
24 ordelon 6090 . . . . . 6 ((Ord dom 𝑂 ∧ (𝑂𝑋) ∈ dom 𝑂) → (𝑂𝑋) ∈ On)
253, 23, 24sylancr 587 . . . . 5 (𝜑 → (𝑂𝑋) ∈ On)
263a1i 11 . . . . . . . 8 (𝜑 → Ord dom 𝑂)
27 ordelon 6090 . . . . . . . 8 ((Ord dom 𝑂 ∧ suc 𝑢 ∈ dom 𝑂) → suc 𝑢 ∈ On)
2826, 27sylan 580 . . . . . . 7 ((𝜑 ∧ suc 𝑢 ∈ dom 𝑂) → suc 𝑢 ∈ On)
29 sucelon 7388 . . . . . . 7 (𝑢 ∈ On ↔ suc 𝑢 ∈ On)
3028, 29sylibr 235 . . . . . 6 ((𝜑 ∧ suc 𝑢 ∈ dom 𝑂) → 𝑢 ∈ On)
3130adantrr 713 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑢 ∈ On)
32 ontri1 6100 . . . . 5 (((𝑂𝑋) ∈ On ∧ 𝑢 ∈ On) → ((𝑂𝑋) ⊆ 𝑢 ↔ ¬ 𝑢 ∈ (𝑂𝑋)))
3325, 31, 32syl2an2r 681 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → ((𝑂𝑋) ⊆ 𝑢 ↔ ¬ 𝑢 ∈ (𝑂𝑋)))
341, 33mpbid 233 . . 3 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → ¬ 𝑢 ∈ (𝑂𝑋))
3512adantr 481 . . . . . 6 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
36 ordtr 6080 . . . . . . . 8 (Ord dom 𝑂 → Tr dom 𝑂)
373, 36mp1i 13 . . . . . . 7 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → Tr dom 𝑂)
38 simprl 767 . . . . . . 7 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → suc 𝑢 ∈ dom 𝑂)
39 trsuc 6150 . . . . . . 7 ((Tr dom 𝑂 ∧ suc 𝑢 ∈ dom 𝑂) → 𝑢 ∈ dom 𝑂)
4037, 38, 39syl2anc 584 . . . . . 6 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑢 ∈ dom 𝑂)
4123adantr 481 . . . . . 6 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑋) ∈ dom 𝑂)
42 isorel 6942 . . . . . 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 833 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑢 E (𝑂𝑋) ↔ (𝑂𝑢) E (𝑂‘(𝑂𝑋))))
44 fvex 6551 . . . . . 6 (𝑂𝑋) ∈ V
4544epeli 5356 . . . . 5 (𝑢 E (𝑂𝑋) ↔ 𝑢 ∈ (𝑂𝑋))
46 fvex 6551 . . . . . 6 (𝑂‘(𝑂𝑋)) ∈ V
4746epeli 5356 . . . . 5 ((𝑂𝑢) E (𝑂‘(𝑂𝑋)) ↔ (𝑂𝑢) ∈ (𝑂‘(𝑂𝑋)))
4843, 45, 473bitr3g 314 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑢 ∈ (𝑂𝑋) ↔ (𝑂𝑢) ∈ (𝑂‘(𝑂𝑋))))
49 f1ocnvfv2 6899 . . . . . . 7 ((𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) ∧ 𝑋 ∈ (𝐺 supp ∅)) → (𝑂‘(𝑂𝑋)) = 𝑋)
5014, 22, 49syl2anc 584 . . . . . 6 (𝜑 → (𝑂‘(𝑂𝑋)) = 𝑋)
5150adantr 481 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂‘(𝑂𝑋)) = 𝑋)
5251eleq2d 2868 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → ((𝑂𝑢) ∈ (𝑂‘(𝑂𝑋)) ↔ (𝑂𝑢) ∈ 𝑋))
5348, 52bitrd 280 . . 3 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑢 ∈ (𝑂𝑋) ↔ (𝑂𝑢) ∈ 𝑋))
5434, 53mtbid 325 . 2 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → ¬ (𝑂𝑢) ∈ 𝑋)
555, 6, 7, 18, 19, 8, 20, 21oemapvali 8993 . . . . 5 (𝜑 → (𝑋𝐵 ∧ (𝐹𝑋) ∈ (𝐺𝑋) ∧ ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤))))
5655simp1d 1135 . . . 4 (𝜑𝑋𝐵)
57 onelon 6091 . . . 4 ((𝐵 ∈ On ∧ 𝑋𝐵) → 𝑋 ∈ On)
587, 56, 57syl2anc 584 . . 3 (𝜑𝑋 ∈ On)
59 suppssdm 7694 . . . . . . 7 (𝐺 supp ∅) ⊆ dom 𝐺
605, 6, 7cantnfs 8975 . . . . . . . . 9 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
618, 60mpbid 233 . . . . . . . 8 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
6261simpld 495 . . . . . . 7 (𝜑𝐺:𝐵𝐴)
6359, 62fssdm 6398 . . . . . 6 (𝜑 → (𝐺 supp ∅) ⊆ 𝐵)
6463adantr 481 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝐺 supp ∅) ⊆ 𝐵)
652oif 8840 . . . . . . 7 𝑂:dom 𝑂⟶(𝐺 supp ∅)
6665ffvelrni 6715 . . . . . 6 (𝑢 ∈ dom 𝑂 → (𝑂𝑢) ∈ (𝐺 supp ∅))
6740, 66syl 17 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑢) ∈ (𝐺 supp ∅))
6864, 67sseldd 3890 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑢) ∈ 𝐵)
69 onelon 6091 . . . 4 ((𝐵 ∈ On ∧ (𝑂𝑢) ∈ 𝐵) → (𝑂𝑢) ∈ On)
707, 68, 69syl2an2r 681 . . 3 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑢) ∈ On)
71 ontri1 6100 . . 3 ((𝑋 ∈ On ∧ (𝑂𝑢) ∈ On) → (𝑋 ⊆ (𝑂𝑢) ↔ ¬ (𝑂𝑢) ∈ 𝑋))
7258, 70, 71syl2an2r 681 . 2 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑋 ⊆ (𝑂𝑢) ↔ ¬ (𝑂𝑢) ∈ 𝑋))
7354, 72mpbird 258 1 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑋 ⊆ (𝑂𝑢))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1522  wcel 2081  wral 3105  wrex 3106  {crab 3109  Vcvv 3437  wss 3859  c0 4211   cuni 4745   class class class wbr 4962  {copab 5024  Tr wtr 5063   E cep 5352   We wwe 5401  ccnv 5442  dom cdm 5443  Ord word 6065  Oncon0 6066  suc csuc 6068  wf 6221  1-1-ontowf1o 6224  cfv 6225   Isom wiso 6226  (class class class)co 7016  ωcom 7436   supp csupp 7681   finSupp cfsupp 8679  OrdIsocoi 8819   CNF ccnf 8970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-supp 7682  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-seqom 7935  df-1o 7953  df-er 8139  df-map 8258  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-fsupp 8680  df-oi 8820  df-cnf 8971
This theorem is referenced by:  cantnflem1c  8996
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