MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nqerid Unicode version

Theorem nqerid 8573
Description: Corollary of nqereu 8569: the function  /Q acts as the identity on members of  Q.. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
nqerid  |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )

Proof of Theorem nqerid
StepHypRef Expression
1 nqerf 8570 . . 3  |-  /Q :
( N.  X.  N. )
--> Q.
2 ffun 5407 . . 3  |-  ( /Q : ( N.  X.  N. ) --> Q.  ->  Fun  /Q )
31, 2ax-mp 8 . 2  |-  Fun  /Q
4 elpqn 8565 . . 3  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
5 id 19 . . 3  |-  ( A  e.  Q.  ->  A  e.  Q. )
6 enqer 8561 . . . . 5  |-  ~Q  Er  ( N.  X.  N. )
76a1i 10 . . . 4  |-  ( A  e.  Q.  ->  ~Q  Er  ( N.  X.  N. )
)
87, 4erref 6696 . . 3  |-  ( A  e.  Q.  ->  A  ~Q  A )
9 df-erq 8553 . . . . 5  |-  /Q  =  (  ~Q  i^i  ( ( N.  X.  N. )  X.  Q. ) )
109breqi 4045 . . . 4  |-  ( A /Q A  <->  A (  ~Q  i^i  ( ( N. 
X.  N. )  X.  Q. ) ) A )
11 brinxp2 4767 . . . 4  |-  ( A (  ~Q  i^i  (
( N.  X.  N. )  X.  Q. ) ) A  <->  ( A  e.  ( N.  X.  N. )  /\  A  e.  Q.  /\  A  ~Q  A ) )
1210, 11bitri 240 . . 3  |-  ( A /Q A  <->  ( A  e.  ( N.  X.  N. )  /\  A  e.  Q.  /\  A  ~Q  A ) )
134, 5, 8, 12syl3anbrc 1136 . 2  |-  ( A  e.  Q.  ->  A /Q A )
14 funbrfv 5577 . 2  |-  ( Fun 
/Q  ->  ( A /Q A  ->  ( /Q `  A )  =  A ) )
153, 13, 14mpsyl 59 1  |-  ( A  e.  Q.  ->  ( /Q `  A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696    i^i cin 3164   class class class wbr 4039    X. cxp 4703   Fun wfun 5265   -->wf 5267   ` cfv 5271    Er wer 6673   N.cnpi 8482    ~Q ceq 8489   Q.cnq 8490   /Qcerq 8492
This theorem is referenced by:  addassnq  8598  mulassnq  8599  distrnq  8601  mulidnq  8603  recmulnq  8604  1lt2nq  8613  ltexnq  8615  prlem934  8673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-ni 8512  df-mi 8514  df-lti 8515  df-enq 8551  df-nq 8552  df-erq 8553  df-1nq 8556
  Copyright terms: Public domain W3C validator