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Theorem cantnff1o 8578
 Description: Simplify the isomorphism of cantnf 8575 to simple bijection. (Contributed by Mario Carneiro, 30-May-2015.)
Hypotheses
Ref Expression
cantnff1o.1 𝑆 = dom (𝐴 CNF 𝐵)
cantnff1o.2 (𝜑𝐴 ∈ On)
cantnff1o.3 (𝜑𝐵 ∈ On)
Assertion
Ref Expression
cantnff1o (𝜑 → (𝐴 CNF 𝐵):𝑆1-1-onto→(𝐴𝑜 𝐵))

Proof of Theorem cantnff1o
Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnff1o.1 . . 3 𝑆 = dom (𝐴 CNF 𝐵)
2 cantnff1o.2 . . 3 (𝜑𝐴 ∈ On)
3 cantnff1o.3 . . 3 (𝜑𝐵 ∈ On)
4 eqid 2620 . . 3 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
51, 2, 3, 4cantnf 8575 . 2 (𝜑 → (𝐴 CNF 𝐵) Isom {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}, E (𝑆, (𝐴𝑜 𝐵)))
6 isof1o 6558 . 2 ((𝐴 CNF 𝐵) Isom {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}, E (𝑆, (𝐴𝑜 𝐵)) → (𝐴 CNF 𝐵):𝑆1-1-onto→(𝐴𝑜 𝐵))
75, 6syl 17 1 (𝜑 → (𝐴 CNF 𝐵):𝑆1-1-onto→(𝐴𝑜 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1481   ∈ wcel 1988  ∀wral 2909  ∃wrex 2910  {copab 4703   E cep 5018  dom cdm 5104  Oncon0 5711  –1-1-onto→wf1o 5875  ‘cfv 5876   Isom wiso 5877  (class class class)co 6635   ↑𝑜 coe 7544   CNF ccnf 8543 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-supp 7281  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-seqom 7528  df-1o 7545  df-2o 7546  df-oadd 7549  df-omul 7550  df-oexp 7551  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-fsupp 8261  df-oi 8400  df-cnf 8544 This theorem is referenced by:  oef1o  8580  cnfcomlem  8581  cnfcom  8582  cnfcom2lem  8583  cnfcom2  8584  cnfcom3lem  8585  cnfcom3  8586
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