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Theorem cantnflem1b 9704
Description: Lemma for cantnf 9711. (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 771 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑋) ⊆ 𝑢)
2 cantnflem1.o . . . . . . 7 𝑂 = OrdIso( E , (𝐺 supp ∅))
32oicl 9547 . . . . . 6 Ord dom 𝑂
4 ovexd 7448 . . . . . . . . . 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 9685 . . . . . . . . . . 11 (𝜑 → ( E We (𝐺 supp ∅) ∧ dom 𝑂 ∈ ω))
109simpld 493 . . . . . . . . . 10 (𝜑 → E We (𝐺 supp ∅))
112oiiso 9555 . . . . . . . . . 10 (((𝐺 supp ∅) ∈ V ∧ E We (𝐺 supp ∅)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
124, 10, 11syl2anc 582 . . . . . . . . 9 (𝜑𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
13 isof1o 7324 . . . . . . . . 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 6844 . . . . . . . 8 (𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) → 𝑂:(𝐺 supp ∅)–1-1-onto→dom 𝑂)
16 f1of 6832 . . . . . . . 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 9703 . . . . . . 7 (𝜑𝑋 ∈ (𝐺 supp ∅))
2317, 22ffvelcdmd 7088 . . . . . 6 (𝜑 → (𝑂𝑋) ∈ dom 𝑂)
24 ordelon 6389 . . . . . 6 ((Ord dom 𝑂 ∧ (𝑂𝑋) ∈ dom 𝑂) → (𝑂𝑋) ∈ On)
253, 23, 24sylancr 585 . . . . 5 (𝜑 → (𝑂𝑋) ∈ On)
263a1i 11 . . . . . . . 8 (𝜑 → Ord dom 𝑂)
27 ordelon 6389 . . . . . . . 8 ((Ord dom 𝑂 ∧ suc 𝑢 ∈ dom 𝑂) → suc 𝑢 ∈ On)
2826, 27sylan 578 . . . . . . 7 ((𝜑 ∧ suc 𝑢 ∈ dom 𝑂) → suc 𝑢 ∈ On)
29 onsucb 7815 . . . . . . 7 (𝑢 ∈ On ↔ suc 𝑢 ∈ On)
3028, 29sylibr 233 . . . . . 6 ((𝜑 ∧ suc 𝑢 ∈ dom 𝑂) → 𝑢 ∈ On)
3130adantrr 715 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑢 ∈ On)
32 ontri1 6399 . . . . 5 (((𝑂𝑋) ∈ On ∧ 𝑢 ∈ On) → ((𝑂𝑋) ⊆ 𝑢 ↔ ¬ 𝑢 ∈ (𝑂𝑋)))
3325, 31, 32syl2an2r 683 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → ((𝑂𝑋) ⊆ 𝑢 ↔ ¬ 𝑢 ∈ (𝑂𝑋)))
341, 33mpbid 231 . . 3 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → ¬ 𝑢 ∈ (𝑂𝑋))
3512adantr 479 . . . . . 6 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑂 Isom E , E (dom 𝑂, (𝐺 supp ∅)))
36 ordtr 6379 . . . . . . . 8 (Ord dom 𝑂 → Tr dom 𝑂)
373, 36mp1i 13 . . . . . . 7 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → Tr dom 𝑂)
38 simprl 769 . . . . . . 7 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → suc 𝑢 ∈ dom 𝑂)
39 trsuc 6452 . . . . . . 7 ((Tr dom 𝑂 ∧ suc 𝑢 ∈ dom 𝑂) → 𝑢 ∈ dom 𝑂)
4037, 38, 39syl2anc 582 . . . . . 6 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑢 ∈ dom 𝑂)
4123adantr 479 . . . . . 6 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑋) ∈ dom 𝑂)
42 isorel 7327 . . . . . 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 835 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑢 E (𝑂𝑋) ↔ (𝑂𝑢) E (𝑂‘(𝑂𝑋))))
44 fvex 6903 . . . . . 6 (𝑂𝑋) ∈ V
4544epeli 5579 . . . . 5 (𝑢 E (𝑂𝑋) ↔ 𝑢 ∈ (𝑂𝑋))
46 fvex 6903 . . . . . 6 (𝑂‘(𝑂𝑋)) ∈ V
4746epeli 5579 . . . . 5 ((𝑂𝑢) E (𝑂‘(𝑂𝑋)) ↔ (𝑂𝑢) ∈ (𝑂‘(𝑂𝑋)))
4843, 45, 473bitr3g 312 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑢 ∈ (𝑂𝑋) ↔ (𝑂𝑢) ∈ (𝑂‘(𝑂𝑋))))
49 f1ocnvfv2 7280 . . . . . . 7 ((𝑂:dom 𝑂1-1-onto→(𝐺 supp ∅) ∧ 𝑋 ∈ (𝐺 supp ∅)) → (𝑂‘(𝑂𝑋)) = 𝑋)
5014, 22, 49syl2anc 582 . . . . . 6 (𝜑 → (𝑂‘(𝑂𝑋)) = 𝑋)
5150adantr 479 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂‘(𝑂𝑋)) = 𝑋)
5251eleq2d 2811 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → ((𝑂𝑢) ∈ (𝑂‘(𝑂𝑋)) ↔ (𝑂𝑢) ∈ 𝑋))
5348, 52bitrd 278 . . 3 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑢 ∈ (𝑂𝑋) ↔ (𝑂𝑢) ∈ 𝑋))
5434, 53mtbid 323 . 2 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → ¬ (𝑂𝑢) ∈ 𝑋)
555, 6, 7, 18, 19, 8, 20, 21oemapvali 9702 . . . . 5 (𝜑 → (𝑋𝐵 ∧ (𝐹𝑋) ∈ (𝐺𝑋) ∧ ∀𝑤𝐵 (𝑋𝑤 → (𝐹𝑤) = (𝐺𝑤))))
5655simp1d 1139 . . . 4 (𝜑𝑋𝐵)
57 onelon 6390 . . . 4 ((𝐵 ∈ On ∧ 𝑋𝐵) → 𝑋 ∈ On)
587, 56, 57syl2anc 582 . . 3 (𝜑𝑋 ∈ On)
59 suppssdm 8175 . . . . . . 7 (𝐺 supp ∅) ⊆ dom 𝐺
605, 6, 7cantnfs 9684 . . . . . . . . 9 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
618, 60mpbid 231 . . . . . . . 8 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
6261simpld 493 . . . . . . 7 (𝜑𝐺:𝐵𝐴)
6359, 62fssdm 6736 . . . . . 6 (𝜑 → (𝐺 supp ∅) ⊆ 𝐵)
6463adantr 479 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝐺 supp ∅) ⊆ 𝐵)
652oif 9548 . . . . . . 7 𝑂:dom 𝑂⟶(𝐺 supp ∅)
6665ffvelcdmi 7086 . . . . . 6 (𝑢 ∈ dom 𝑂 → (𝑂𝑢) ∈ (𝐺 supp ∅))
6740, 66syl 17 . . . . 5 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑢) ∈ (𝐺 supp ∅))
6864, 67sseldd 3974 . . . 4 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑢) ∈ 𝐵)
69 onelon 6390 . . . 4 ((𝐵 ∈ On ∧ (𝑂𝑢) ∈ 𝐵) → (𝑂𝑢) ∈ On)
707, 68, 69syl2an2r 683 . . 3 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑂𝑢) ∈ On)
71 ontri1 6399 . . 3 ((𝑋 ∈ On ∧ (𝑂𝑢) ∈ On) → (𝑋 ⊆ (𝑂𝑢) ↔ ¬ (𝑂𝑢) ∈ 𝑋))
7258, 70, 71syl2an2r 683 . 2 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → (𝑋 ⊆ (𝑂𝑢) ↔ ¬ (𝑂𝑢) ∈ 𝑋))
7354, 72mpbird 256 1 ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (𝑂𝑋) ⊆ 𝑢)) → 𝑋 ⊆ (𝑂𝑢))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3051  wrex 3060  {crab 3419  Vcvv 3463  wss 3941  c0 4319   cuni 4904   class class class wbr 5144  {copab 5206  Tr wtr 5261   E cep 5576   We wwe 5627  ccnv 5672  dom cdm 5673  Ord word 6364  Oncon0 6365  suc csuc 6367  wf 6539  1-1-ontowf1o 6542  cfv 6543   Isom wiso 6544  (class class class)co 7413  ωcom 7865   supp csupp 8158   finSupp cfsupp 9380  OrdIsocoi 9527   CNF ccnf 9679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-se 5629  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-2nd 7988  df-supp 8159  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-seqom 8462  df-1o 8480  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fsupp 9381  df-oi 9528  df-cnf 9680
This theorem is referenced by:  cantnflem1c  9705
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