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Theorem cantnffval2 9736
Description: An alternate definition of df-cnf 9703 which relies on cantnf 9734. (Note that although the use of 𝑆 seems self-referential, one can use cantnfdm 9705 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 9734 . . . 4 (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴o 𝐵)))
6 isof1o 7344 . . . 4 ((𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴o 𝐵)) → (𝐴 CNF 𝐵):𝑆1-1-onto→(𝐴o 𝐵))
7 f1orel 6850 . . . 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 6208 . . 3 (Rel (𝐴 CNF 𝐵) ↔ (𝐴 CNF 𝐵) = (𝐴 CNF 𝐵))
108, 9sylib 218 . 2 (𝜑(𝐴 CNF 𝐵) = (𝐴 CNF 𝐵))
11 oecl 8576 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
122, 3, 11syl2anc 584 . . . . . 6 (𝜑 → (𝐴o 𝐵) ∈ On)
13 eloni 6393 . . . . . 6 ((𝐴o 𝐵) ∈ On → Ord (𝐴o 𝐵))
1412, 13syl 17 . . . . 5 (𝜑 → Ord (𝐴o 𝐵))
15 isocnv 7351 . . . . . 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 9735 . . . . . . 7 (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 𝑆) = (𝐴o 𝐵)))
1817simpld 494 . . . . . 6 (𝜑𝑇 We 𝑆)
19 ovex 7465 . . . . . . . . 9 (𝐴 CNF 𝐵) ∈ V
2019dmex 7932 . . . . . . . 8 dom (𝐴 CNF 𝐵) ∈ V
211, 20eqeltri 2836 . . . . . . 7 𝑆 ∈ V
22 exse 5644 . . . . . . 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 9580 . . . . . 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 5885 . 2 (𝜑(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))
3010, 29eqtr3d 2778 1 (𝜑 → (𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wral 3060  wrex 3069  Vcvv 3479  {copab 5204   E cep 5582   Se wse 5634   We wwe 5635  ccnv 5683  dom cdm 5684  Rel wrel 5689  Ord word 6382  Oncon0 6383  1-1-ontowf1o 6559  cfv 6560   Isom wiso 6561  (class class class)co 7432  o coe 8506  OrdIsocoi 9550   CNF ccnf 9702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-seqom 8489  df-1o 8507  df-2o 8508  df-oadd 8511  df-omul 8512  df-oexp 8513  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-oi 9551  df-cnf 9703
This theorem is referenced by: (None)
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