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Theorem nqerrel 9960
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 9958 . . . . 5 [Q]:(N × N)⟶Q
3 ffn 6184 . . . . 5 ([Q]:(N × N)⟶Q → [Q] Fn (N × N))
42, 3ax-mp 5 . . . 4 [Q] Fn (N × N)
5 fnbrfvb 6379 . . . 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 9941 . . . 4 [Q] = ( ~Q ∩ ((N × N) × Q))
9 inss1 3981 . . . 4 ( ~Q ∩ ((N × N) × Q)) ⊆ ~Q
108, 9eqsstri 3784 . . 3 [Q] ⊆ ~Q
1110ssbri 4832 . 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 4787   × cxp 5248   Fn wfn 6025  wf 6026  cfv 6030  Ncnpi 9872   ~Q ceq 9879  Qcnq 9880  [Q]cerq 9882
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 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
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 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-omul 7722  df-er 7900  df-ni 9900  df-mi 9902  df-lti 9903  df-enq 9939  df-nq 9940  df-erq 9941  df-1nq 9944
This theorem is referenced by:  nqereq  9963  adderpq  9984  mulerpq  9985  lterpq  9998
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