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Theorem nqerrel 9739
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 2620 . . 3 ([Q]‘𝐴) = ([Q]‘𝐴)
2 nqerf 9737 . . . . 5 [Q]:(N × N)⟶Q
3 ffn 6032 . . . . 5 ([Q]:(N × N)⟶Q → [Q] Fn (N × N))
42, 3ax-mp 5 . . . 4 [Q] Fn (N × N)
5 fnbrfvb 6223 . . . 4 (([Q] Fn (N × N) ∧ 𝐴 ∈ (N × N)) → (([Q]‘𝐴) = ([Q]‘𝐴) ↔ 𝐴[Q]([Q]‘𝐴)))
64, 5mpan 705 . . 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 9720 . . . 4 [Q] = ( ~Q ∩ ((N × N) × Q))
9 inss1 3825 . . . 4 ( ~Q ∩ ((N × N) × Q)) ⊆ ~Q
108, 9eqsstri 3627 . . 3 [Q] ⊆ ~Q
1110ssbri 4688 . 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 1481  wcel 1988  cin 3566   class class class wbr 4644   × cxp 5102   Fn wfn 5871  wf 5872  cfv 5876  Ncnpi 9651   ~Q ceq 9658  Qcnq 9659  [Q]cerq 9661
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-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-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-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-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-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-omul 7550  df-er 7727  df-ni 9679  df-mi 9681  df-lti 9682  df-enq 9718  df-nq 9719  df-erq 9720  df-1nq 9723
This theorem is referenced by:  nqereq  9742  adderpq  9763  mulerpq  9764  lterpq  9777
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