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Theorem cantnffval2 8536
Description: An alternate definition of df-cnf 8503 which relies on cantnf 8534. (Note that although the use of 𝑆 seems self-referential, one can use cantnfdm 8505 to eliminate it.) (Contributed by Mario Carneiro, 28-May-2015.)
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)))}
Assertion
Ref Expression
cantnffval2 (𝜑 → (𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐵   𝑤,𝐴,𝑥,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑤)   𝑆(𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cantnffval2
StepHypRef Expression
1 cantnfs.s . . . . 5 𝑆 = dom (𝐴 CNF 𝐵)
2 cantnfs.a . . . . 5 (𝜑𝐴 ∈ On)
3 cantnfs.b . . . . 5 (𝜑𝐵 ∈ On)
4 oemapval.t . . . . 5 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
51, 2, 3, 4cantnf 8534 . . . 4 (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴𝑜 𝐵)))
6 isof1o 6527 . . . 4 ((𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴𝑜 𝐵)) → (𝐴 CNF 𝐵):𝑆1-1-onto→(𝐴𝑜 𝐵))
7 f1orel 6097 . . . 4 ((𝐴 CNF 𝐵):𝑆1-1-onto→(𝐴𝑜 𝐵) → Rel (𝐴 CNF 𝐵))
85, 6, 73syl 18 . . 3 (𝜑 → Rel (𝐴 CNF 𝐵))
9 dfrel2 5542 . . 3 (Rel (𝐴 CNF 𝐵) ↔ (𝐴 CNF 𝐵) = (𝐴 CNF 𝐵))
108, 9sylib 208 . 2 (𝜑(𝐴 CNF 𝐵) = (𝐴 CNF 𝐵))
11 oecl 7562 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
122, 3, 11syl2anc 692 . . . . . 6 (𝜑 → (𝐴𝑜 𝐵) ∈ On)
13 eloni 5692 . . . . . 6 ((𝐴𝑜 𝐵) ∈ On → Ord (𝐴𝑜 𝐵))
1412, 13syl 17 . . . . 5 (𝜑 → Ord (𝐴𝑜 𝐵))
15 isocnv 6534 . . . . . 6 ((𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴𝑜 𝐵)) → (𝐴 CNF 𝐵) Isom E , 𝑇 ((𝐴𝑜 𝐵), 𝑆))
165, 15syl 17 . . . . 5 (𝜑(𝐴 CNF 𝐵) Isom E , 𝑇 ((𝐴𝑜 𝐵), 𝑆))
171, 2, 3, 4oemapwe 8535 . . . . . . 7 (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 𝑆) = (𝐴𝑜 𝐵)))
1817simpld 475 . . . . . 6 (𝜑𝑇 We 𝑆)
19 ovex 6632 . . . . . . . . 9 (𝐴 CNF 𝐵) ∈ V
2019dmex 7046 . . . . . . . 8 dom (𝐴 CNF 𝐵) ∈ V
211, 20eqeltri 2694 . . . . . . 7 𝑆 ∈ V
22 exse 5038 . . . . . . 7 (𝑆 ∈ V → 𝑇 Se 𝑆)
2321, 22ax-mp 5 . . . . . 6 𝑇 Se 𝑆
24 eqid 2621 . . . . . . 7 OrdIso(𝑇, 𝑆) = OrdIso(𝑇, 𝑆)
2524oieu 8388 . . . . . 6 ((𝑇 We 𝑆𝑇 Se 𝑆) → ((Ord (𝐴𝑜 𝐵) ∧ (𝐴 CNF 𝐵) Isom E , 𝑇 ((𝐴𝑜 𝐵), 𝑆)) ↔ ((𝐴𝑜 𝐵) = dom OrdIso(𝑇, 𝑆) ∧ (𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))))
2618, 23, 25sylancl 693 . . . . 5 (𝜑 → ((Ord (𝐴𝑜 𝐵) ∧ (𝐴 CNF 𝐵) Isom E , 𝑇 ((𝐴𝑜 𝐵), 𝑆)) ↔ ((𝐴𝑜 𝐵) = dom OrdIso(𝑇, 𝑆) ∧ (𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))))
2714, 16, 26mpbi2and 955 . . . 4 (𝜑 → ((𝐴𝑜 𝐵) = dom OrdIso(𝑇, 𝑆) ∧ (𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆)))
2827simprd 479 . . 3 (𝜑(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))
2928cnveqd 5258 . 2 (𝜑(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))
3010, 29eqtr3d 2657 1 (𝜑 → (𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  {copab 4672   E cep 4983   Se wse 5031   We wwe 5032  ccnv 5073  dom cdm 5074  Rel wrel 5079  Ord word 5681  Oncon0 5682  1-1-ontowf1o 5846  cfv 5847   Isom wiso 5848  (class class class)co 6604  𝑜 coe 7504  OrdIsocoi 8358   CNF ccnf 8502
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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-seqom 7488  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510  df-oexp 7511  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-oi 8359  df-cnf 8503
This theorem is referenced by: (None)
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