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Theorem cantnffval2 9604
Description: An alternate definition of df-cnf 9571 which relies on cantnf 9602. (Note that although the use of 𝑆 seems self-referential, one can use cantnfdm 9573 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 9602 . . . 4 (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴o 𝐵)))
6 isof1o 7269 . . . 4 ((𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴o 𝐵)) → (𝐴 CNF 𝐵):𝑆1-1-onto→(𝐴o 𝐵))
7 f1orel 6777 . . . 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 6147 . . 3 (Rel (𝐴 CNF 𝐵) ↔ (𝐴 CNF 𝐵) = (𝐴 CNF 𝐵))
108, 9sylib 218 . 2 (𝜑(𝐴 CNF 𝐵) = (𝐴 CNF 𝐵))
11 oecl 8464 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
122, 3, 11syl2anc 584 . . . . . 6 (𝜑 → (𝐴o 𝐵) ∈ On)
13 eloni 6327 . . . . . 6 ((𝐴o 𝐵) ∈ On → Ord (𝐴o 𝐵))
1412, 13syl 17 . . . . 5 (𝜑 → Ord (𝐴o 𝐵))
15 isocnv 7276 . . . . . 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 9603 . . . . . . 7 (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 𝑆) = (𝐴o 𝐵)))
1817simpld 494 . . . . . 6 (𝜑𝑇 We 𝑆)
19 ovex 7391 . . . . . . . . 9 (𝐴 CNF 𝐵) ∈ V
2019dmex 7851 . . . . . . . 8 dom (𝐴 CNF 𝐵) ∈ V
211, 20eqeltri 2832 . . . . . . 7 𝑆 ∈ V
22 exse 5584 . . . . . . 7 (𝑆 ∈ V → 𝑇 Se 𝑆)
2321, 22ax-mp 5 . . . . . 6 𝑇 Se 𝑆
24 eqid 2736 . . . . . . 7 OrdIso(𝑇, 𝑆) = OrdIso(𝑇, 𝑆)
2524oieu 9444 . . . . . 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 586 . . . . 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 712 . . . 4 (𝜑 → ((𝐴o 𝐵) = dom OrdIso(𝑇, 𝑆) ∧ (𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆)))
2827simprd 495 . . 3 (𝜑(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))
2928cnveqd 5824 . 2 (𝜑(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))
3010, 29eqtr3d 2773 1 (𝜑 → (𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1541  wcel 2113  wral 3051  wrex 3060  Vcvv 3440  {copab 5160   E cep 5523   Se wse 5575   We wwe 5576  ccnv 5623  dom cdm 5624  Rel wrel 5629  Ord word 6316  Oncon0 6317  1-1-ontowf1o 6491  cfv 6492   Isom wiso 6493  (class class class)co 7358  o coe 8396  OrdIsocoi 9414   CNF ccnf 9570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-seqom 8379  df-1o 8397  df-2o 8398  df-oadd 8401  df-omul 8402  df-oexp 8403  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-oi 9415  df-cnf 9571
This theorem is referenced by: (None)
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