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Theorem cantnffval2 9726
Description: An alternate definition of df-cnf 9693 which relies on cantnf 9724. (Note that although the use of 𝑆 seems self-referential, one can use cantnfdm 9695 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 9724 . . . 4 (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴o 𝐵)))
6 isof1o 7337 . . . 4 ((𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴o 𝐵)) → (𝐴 CNF 𝐵):𝑆1-1-onto→(𝐴o 𝐵))
7 f1orel 6847 . . . 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 6198 . . 3 (Rel (𝐴 CNF 𝐵) ↔ (𝐴 CNF 𝐵) = (𝐴 CNF 𝐵))
108, 9sylib 217 . 2 (𝜑(𝐴 CNF 𝐵) = (𝐴 CNF 𝐵))
11 oecl 8564 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
122, 3, 11syl2anc 582 . . . . . 6 (𝜑 → (𝐴o 𝐵) ∈ On)
13 eloni 6384 . . . . . 6 ((𝐴o 𝐵) ∈ On → Ord (𝐴o 𝐵))
1412, 13syl 17 . . . . 5 (𝜑 → Ord (𝐴o 𝐵))
15 isocnv 7344 . . . . . 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 9725 . . . . . . 7 (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 𝑆) = (𝐴o 𝐵)))
1817simpld 493 . . . . . 6 (𝜑𝑇 We 𝑆)
19 ovex 7459 . . . . . . . . 9 (𝐴 CNF 𝐵) ∈ V
2019dmex 7923 . . . . . . . 8 dom (𝐴 CNF 𝐵) ∈ V
211, 20eqeltri 2825 . . . . . . 7 𝑆 ∈ V
22 exse 5645 . . . . . . 7 (𝑆 ∈ V → 𝑇 Se 𝑆)
2321, 22ax-mp 5 . . . . . 6 𝑇 Se 𝑆
24 eqid 2728 . . . . . . 7 OrdIso(𝑇, 𝑆) = OrdIso(𝑇, 𝑆)
2524oieu 9570 . . . . . 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 584 . . . . 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 710 . . . 4 (𝜑 → ((𝐴o 𝐵) = dom OrdIso(𝑇, 𝑆) ∧ (𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆)))
2827simprd 494 . . 3 (𝜑(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))
2928cnveqd 5882 . 2 (𝜑(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))
3010, 29eqtr3d 2770 1 (𝜑 → (𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3058  wrex 3067  Vcvv 3473  {copab 5214   E cep 5585   Se wse 5635   We wwe 5636  ccnv 5681  dom cdm 5682  Rel wrel 5687  Ord word 6373  Oncon0 6374  1-1-ontowf1o 6552  cfv 6553   Isom wiso 6554  (class class class)co 7426  o coe 8492  OrdIsocoi 9540   CNF ccnf 9692
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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-supp 8172  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-seqom 8475  df-1o 8493  df-2o 8494  df-oadd 8497  df-omul 8498  df-oexp 8499  df-er 8731  df-map 8853  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-fsupp 9394  df-oi 9541  df-cnf 9693
This theorem is referenced by: (None)
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