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Theorem nqerrel 9357
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 2422 . . 3  |-  ( /Q
`  A )  =  ( /Q `  A
)
2 nqerf 9355 . . . . 5  |-  /Q :
( N.  X.  N. )
--> Q.
3 ffn 5742 . . . . 5  |-  ( /Q : ( N.  X.  N. ) --> Q.  ->  /Q  Fn  ( N.  X.  N. )
)
42, 3ax-mp 5 . . . 4  |-  /Q  Fn  ( N.  X.  N. )
5 fnbrfvb 5917 . . . 4  |-  ( ( /Q  Fn  ( N. 
X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  (
( /Q `  A
)  =  ( /Q
`  A )  <->  A /Q ( /Q `  A ) ) )
64, 5mpan 674 . . 3  |-  ( A  e.  ( N.  X.  N. )  ->  ( ( /Q `  A )  =  ( /Q `  A )  <->  A /Q ( /Q `  A ) ) )
71, 6mpbii 214 . 2  |-  ( A  e.  ( N.  X.  N. )  ->  A /Q ( /Q `  A ) )
8 df-erq 9338 . . . 4  |-  /Q  =  (  ~Q  i^i  ( ( N.  X.  N. )  X.  Q. ) )
9 inss1 3682 . . . 4  |-  (  ~Q  i^i  ( ( N.  X.  N. )  X.  Q. )
)  C_  ~Q
108, 9eqsstri 3494 . . 3  |-  /Q  C_  ~Q
1110ssbri 4463 . 2  |-  ( A /Q ( /Q `  A )  ->  A  ~Q  ( /Q `  A
) )
127, 11syl 17 1  |-  ( A  e.  ( N.  X.  N. )  ->  A  ~Q  ( /Q `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437    e. wcel 1868    i^i cin 3435   class class class wbr 4420    X. cxp 4847    Fn wfn 5592   -->wf 5593   ` cfv 5597   N.cnpi 9269    ~Q ceq 9276   Q.cnq 9277   /Qcerq 9279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-oadd 7190  df-omul 7191  df-er 7367  df-ni 9297  df-mi 9299  df-lti 9300  df-enq 9336  df-nq 9337  df-erq 9338  df-1nq 9341
This theorem is referenced by:  nqereq  9360  adderpq  9381  mulerpq  9382  lterpq  9395
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