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Theorem nqerrel 9122
Description: Any member of  ( N. 
X.  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  e.  ( N.  X.  N. )  ->  A  ~Q  ( /Q `  A ) )

Proof of Theorem nqerrel
StepHypRef Expression
1 eqid 2443 . . 3  |-  ( /Q
`  A )  =  ( /Q `  A
)
2 nqerf 9120 . . . . 5  |-  /Q :
( N.  X.  N. )
--> Q.
3 ffn 5580 . . . . 5  |-  ( /Q : ( N.  X.  N. ) --> Q.  ->  /Q  Fn  ( N.  X.  N. )
)
42, 3ax-mp 5 . . . 4  |-  /Q  Fn  ( N.  X.  N. )
5 fnbrfvb 5753 . . . 4  |-  ( ( /Q  Fn  ( N. 
X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  (
( /Q `  A
)  =  ( /Q
`  A )  <->  A /Q ( /Q `  A ) ) )
64, 5mpan 670 . . 3  |-  ( A  e.  ( N.  X.  N. )  ->  ( ( /Q `  A )  =  ( /Q `  A )  <->  A /Q ( /Q `  A ) ) )
71, 6mpbii 211 . 2  |-  ( A  e.  ( N.  X.  N. )  ->  A /Q ( /Q `  A ) )
8 df-erq 9103 . . . 4  |-  /Q  =  (  ~Q  i^i  ( ( N.  X.  N. )  X.  Q. ) )
9 inss1 3591 . . . 4  |-  (  ~Q  i^i  ( ( N.  X.  N. )  X.  Q. )
)  C_  ~Q
108, 9eqsstri 3407 . . 3  |-  /Q  C_  ~Q
1110ssbri 4355 . 2  |-  ( A /Q ( /Q `  A )  ->  A  ~Q  ( /Q `  A
) )
127, 11syl 16 1  |-  ( A  e.  ( N.  X.  N. )  ->  A  ~Q  ( /Q `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756    i^i cin 3348   class class class wbr 4313    X. cxp 4859    Fn wfn 5434   -->wf 5435   ` cfv 5439   N.cnpi 9032    ~Q ceq 9039   Q.cnq 9040   /Qcerq 9042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-omul 6946  df-er 7122  df-ni 9062  df-mi 9064  df-lti 9065  df-enq 9101  df-nq 9102  df-erq 9103  df-1nq 9106
This theorem is referenced by:  nqereq  9125  adderpq  9146  mulerpq  9147  lterpq  9160
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