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Theorem cantnflem1b 9755
Description: Lemma for cantnf 9762. (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 772 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑋) ⊆ 𝑢)
2 cantnflem1.o . . . . . . 7 𝑂 = OrdIso( E , (𝐺 supp ∅))
32oicl 9598 . . . . . 6 Ord dom 𝑂
4 ovexd 7483 . . . . . . . . . 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 9736 . . . . . . . . . . 11 (𝜑 → ( E We (𝐺 supp ∅) ∧ dom 𝑂 ∈ ω))
109simpld 494 . . . . . . . . . 10 (𝜑 → E We (𝐺 supp ∅))
112oiiso 9606 . . . . . . . . . 10 (((𝐺 supp ∅) ∈ V ∧ E We (𝐺 supp ∅)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
124, 10, 11syl2anc 583 . . . . . . . . 9 (𝜑𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
13 isof1o 7359 . . . . . . . . 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 6874 . . . . . . . 8 (𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) → 𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂)
16 f1of 6862 . . . . . . . 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 9754 . . . . . . 7 (𝜑𝑋 ∈ (𝐺 supp ∅))
2317, 22ffvelcdmd 7119 . . . . . 6 (𝜑 → (𝑂𝑋) ∈ dom 𝑂)
24 ordelon 6419 . . . . . 6 ((Ord dom 𝑂 ∧ (𝑂𝑋) ∈ dom 𝑂) → (𝑂𝑋) ∈ On)
253, 23, 24sylancr 586 . . . . 5 (𝜑 → (𝑂𝑋) ∈ On)
263a1i 11 . . . . . . . 8 (𝜑 → Ord dom 𝑂)
27 ordelon 6419 . . . . . . . 8 ((Ord dom 𝑂 ∧ suc 𝑢 ∈ dom 𝑂) → suc 𝑢 ∈ On)
2826, 27sylan 579 . . . . . . 7 ((𝜑 ∧ suc 𝑢 ∈ dom 𝑂) → suc 𝑢 ∈ On)
29 onsucb 7853 . . . . . . 7 (𝑢 ∈ On ↔ suc 𝑢 ∈ On)
3028, 29sylibr 234 . . . . . 6 ((𝜑 ∧ suc 𝑢 ∈ dom 𝑂) → 𝑢 ∈ On)
3130adantrr 716 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑢 ∈ On)
32 ontri1 6429 . . . . 5 (((𝑂𝑋) ∈ On ∧ 𝑢 ∈ On) → ((𝑂𝑋) ⊆ 𝑢 ↔ ¬ 𝑢 ∈ (𝑂𝑋)))
3325, 31, 32syl2an2r 684 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → ((𝑂𝑋) ⊆ 𝑢 ↔ ¬ 𝑢 ∈ (𝑂𝑋)))
341, 33mpbid 232 . . 3 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → ¬ 𝑢 ∈ (𝑂𝑋))
3512adantr 480 . . . . . 6 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
36 ordtr 6409 . . . . . . . 8 (Ord dom 𝑂 → Tr dom 𝑂)
373, 36mp1i 13 . . . . . . 7 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → Tr dom 𝑂)
38 simprl 770 . . . . . . 7 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → suc 𝑢 ∈ dom 𝑂)
39 trsuc 6482 . . . . . . 7 ((Tr dom 𝑂 ∧ suc 𝑢 ∈ dom 𝑂) → 𝑢 ∈ dom 𝑂)
4037, 38, 39syl2anc 583 . . . . . 6 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑢 ∈ dom 𝑂)
4123adantr 480 . . . . . 6 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑋) ∈ dom 𝑂)
42 isorel 7362 . . . . . 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 836 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑢 E (𝑂𝑋) ↔ (𝑂𝑢) E (𝑂‘(𝑂𝑋))))
44 fvex 6933 . . . . . 6 (𝑂𝑋) ∈ V
4544epeli 5601 . . . . 5 (𝑢 E (𝑂𝑋) ↔ 𝑢 ∈ (𝑂𝑋))
46 fvex 6933 . . . . . 6 (𝑂‘(𝑂𝑋)) ∈ V
4746epeli 5601 . . . . 5 ((𝑂𝑢) E (𝑂‘(𝑂𝑋)) ↔ (𝑂𝑢) ∈ (𝑂‘(𝑂𝑋)))
4843, 45, 473bitr3g 313 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑢 ∈ (𝑂𝑋) ↔ (𝑂𝑢) ∈ (𝑂‘(𝑂𝑋))))
49 f1ocnvfv2 7313 . . . . . . 7 ((𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) ∧ 𝑋 ∈ (𝐺 supp ∅)) → (𝑂‘(𝑂𝑋)) = 𝑋)
5014, 22, 49syl2anc 583 . . . . . 6 (𝜑 → (𝑂‘(𝑂𝑋)) = 𝑋)
5150adantr 480 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂‘(𝑂𝑋)) = 𝑋)
5251eleq2d 2830 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → ((𝑂𝑢) ∈ (𝑂‘(𝑂𝑋)) ↔ (𝑂𝑢) ∈ 𝑋))
5348, 52bitrd 279 . . 3 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑢 ∈ (𝑂𝑋) ↔ (𝑂𝑢) ∈ 𝑋))
5434, 53mtbid 324 . 2 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → ¬ (𝑂𝑢) ∈ 𝑋)
555, 6, 7, 18, 19, 8, 20, 21oemapvali 9753 . . . . 5 (𝜑 → (𝑋𝐵 ∧ (𝐹𝑋) ∈ (𝐺𝑋) ∧ ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤))))
5655simp1d 1142 . . . 4 (𝜑𝑋𝐵)
57 onelon 6420 . . . 4 ((𝐵 ∈ On ∧ 𝑋𝐵) → 𝑋 ∈ On)
587, 56, 57syl2anc 583 . . 3 (𝜑𝑋 ∈ On)
59 suppssdm 8218 . . . . . . 7 (𝐺 supp ∅) ⊆ dom 𝐺
605, 6, 7cantnfs 9735 . . . . . . . . 9 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
618, 60mpbid 232 . . . . . . . 8 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
6261simpld 494 . . . . . . 7 (𝜑𝐺:𝐵𝐴)
6359, 62fssdm 6766 . . . . . 6 (𝜑 → (𝐺 supp ∅) ⊆ 𝐵)
6463adantr 480 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝐺 supp ∅) ⊆ 𝐵)
652oif 9599 . . . . . . 7 𝑂:dom 𝑂⟶(𝐺 supp ∅)
6665ffvelcdmi 7117 . . . . . 6 (𝑢 ∈ dom 𝑂 → (𝑂𝑢) ∈ (𝐺 supp ∅))
6740, 66syl 17 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑢) ∈ (𝐺 supp ∅))
6864, 67sseldd 4009 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑢) ∈ 𝐵)
69 onelon 6420 . . . 4 ((𝐵 ∈ On ∧ (𝑂𝑢) ∈ 𝐵) → (𝑂𝑢) ∈ On)
707, 68, 69syl2an2r 684 . . 3 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑢) ∈ On)
71 ontri1 6429 . . 3 ((𝑋 ∈ On ∧ (𝑂𝑢) ∈ On) → (𝑋 ⊆ (𝑂𝑢) ↔ ¬ (𝑂𝑢) ∈ 𝑋))
7258, 70, 71syl2an2r 684 . 2 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑋 ⊆ (𝑂𝑢) ↔ ¬ (𝑂𝑢) ∈ 𝑋))
7354, 72mpbird 257 1 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑋 ⊆ (𝑂𝑢))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2108  wral 3067  wrex 3076  {crab 3443  Vcvv 3488  wss 3976  c0 4352   cuni 4931   class class class wbr 5166  {copab 5228  Tr wtr 5283   E cep 5598   We wwe 5651  ccnv 5699  dom cdm 5700  Ord word 6394  Oncon0 6395  suc csuc 6397  wf 6569  1-1-ontowf1o 6572  cfv 6573   Isom wiso 6574  (class class class)co 7448  ωcom 7903   supp csupp 8201   finSupp cfsupp 9431  OrdIsocoi 9578   CNF ccnf 9730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-supp 8202  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-seqom 8504  df-1o 8522  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-oi 9579  df-cnf 9731
This theorem is referenced by:  cantnflem1c  9756
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