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Theorem nqerrel 9956
Description: Any member of (N × N) relates to the representative of its equivalence class. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
nqerrel (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴))

Proof of Theorem nqerrel
StepHypRef Expression
1 eqid 2771 . . 3 ([Q]‘𝐴) = ([Q]‘𝐴)
2 nqerf 9954 . . . . 5 [Q]:(N × N)⟶Q
3 ffn 6185 . . . . 5 ([Q]:(N × N)⟶Q → [Q] Fn (N × N))
42, 3ax-mp 5 . . . 4 [Q] Fn (N × N)
5 fnbrfvb 6377 . . . 4 (([Q] Fn (N × N) ∧ 𝐴 ∈ (N × N)) → (([Q]‘𝐴) = ([Q]‘𝐴) ↔ 𝐴[Q]([Q]‘𝐴)))
64, 5mpan 670 . . 3 (𝐴 ∈ (N × N) → (([Q]‘𝐴) = ([Q]‘𝐴) ↔ 𝐴[Q]([Q]‘𝐴)))
71, 6mpbii 223 . 2 (𝐴 ∈ (N × N) → 𝐴[Q]([Q]‘𝐴))
8 df-erq 9937 . . . 4 [Q] = ( ~Q ∩ ((N × N) × Q))
9 inss1 3981 . . . 4 ( ~Q ∩ ((N × N) × Q)) ⊆ ~Q
108, 9eqsstri 3784 . . 3 [Q] ⊆ ~Q
1110ssbri 4831 . 2 (𝐴[Q]([Q]‘𝐴) → 𝐴 ~Q ([Q]‘𝐴))
127, 11syl 17 1 (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  cin 3722   class class class wbr 4786   × cxp 5247   Fn wfn 6026  wf 6027  cfv 6031  Ncnpi 9868   ~Q ceq 9875  Qcnq 9876  [Q]cerq 9878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-omul 7718  df-er 7896  df-ni 9896  df-mi 9898  df-lti 9899  df-enq 9935  df-nq 9936  df-erq 9937  df-1nq 9940
This theorem is referenced by:  nqereq  9959  adderpq  9980  mulerpq  9981  lterpq  9994
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