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Theorem cantnffval2 8475
 Description: An alternate definition of df-cnf 8442 which relies on cantnf 8473. (Note that although the use of 𝑆 seems self-referential, one can use cantnfdm 8444 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 8473 . . . 4 (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴𝑜 𝐵)))
6 isof1o 6473 . . . 4 ((𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴𝑜 𝐵)) → (𝐴 CNF 𝐵):𝑆1-1-onto→(𝐴𝑜 𝐵))
7 f1orel 6053 . . . 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 5502 . . 3 (Rel (𝐴 CNF 𝐵) ↔ (𝐴 CNF 𝐵) = (𝐴 CNF 𝐵))
108, 9sylib 207 . 2 (𝜑(𝐴 CNF 𝐵) = (𝐴 CNF 𝐵))
11 oecl 7504 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
122, 3, 11syl2anc 691 . . . . . 6 (𝜑 → (𝐴𝑜 𝐵) ∈ On)
13 eloni 5650 . . . . . 6 ((𝐴𝑜 𝐵) ∈ On → Ord (𝐴𝑜 𝐵))
1412, 13syl 17 . . . . 5 (𝜑 → Ord (𝐴𝑜 𝐵))
15 isocnv 6480 . . . . . 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 8474 . . . . . . 7 (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 𝑆) = (𝐴𝑜 𝐵)))
1817simpld 474 . . . . . 6 (𝜑𝑇 We 𝑆)
19 ovex 6577 . . . . . . . . 9 (𝐴 CNF 𝐵) ∈ V
2019dmex 6991 . . . . . . . 8 dom (𝐴 CNF 𝐵) ∈ V
211, 20eqeltri 2684 . . . . . . 7 𝑆 ∈ V
22 exse 5002 . . . . . . 7 (𝑆 ∈ V → 𝑇 Se 𝑆)
2321, 22ax-mp 5 . . . . . 6 𝑇 Se 𝑆
24 eqid 2610 . . . . . . 7 OrdIso(𝑇, 𝑆) = OrdIso(𝑇, 𝑆)
2524oieu 8327 . . . . . 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 958 . . . 4 (𝜑 → ((𝐴𝑜 𝐵) = dom OrdIso(𝑇, 𝑆) ∧ (𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆)))
2827simprd 478 . . 3 (𝜑(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))
2928cnveqd 5220 . 2 (𝜑(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))
3010, 29eqtr3d 2646 1 (𝜑 → (𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173  {copab 4642   E cep 4947   Se wse 4995   We wwe 4996  ◡ccnv 5037  dom cdm 5038  Rel wrel 5043  Ord word 5639  Oncon0 5640  –1-1-onto→wf1o 5803  ‘cfv 5804   Isom wiso 5805  (class class class)co 6549   ↑𝑜 coe 7446  OrdIsocoi 8297   CNF ccnf 8441 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-cnf 8442 This theorem is referenced by: (None)
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