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

Theorem fovcl 6297
Description: Closure law for an operation. (Contributed by NM, 19-Apr-2007.)
Hypothesis
Ref Expression
fovcl.1  |-  F :
( R  X.  S
) --> C
Assertion
Ref Expression
fovcl  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( A F B )  e.  C )

Proof of Theorem fovcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fovcl.1 . . 3  |-  F :
( R  X.  S
) --> C
2 ffnov 6296 . . . 4  |-  ( F : ( R  X.  S ) --> C  <->  ( F  Fn  ( R  X.  S
)  /\  A. x  e.  R  A. y  e.  S  ( x F y )  e.  C ) )
32simprbi 464 . . 3  |-  ( F : ( R  X.  S ) --> C  ->  A. x  e.  R  A. y  e.  S  ( x F y )  e.  C )
41, 3ax-mp 5 . 2  |-  A. x  e.  R  A. y  e.  S  ( x F y )  e.  C
5 oveq1 6199 . . . 4  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
65eleq1d 2520 . . 3  |-  ( x  =  A  ->  (
( x F y )  e.  C  <->  ( A F y )  e.  C ) )
7 oveq2 6200 . . . 4  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
87eleq1d 2520 . . 3  |-  ( y  =  B  ->  (
( A F y )  e.  C  <->  ( A F B )  e.  C
) )
96, 8rspc2v 3178 . 2  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( A. x  e.  R  A. y  e.  S  ( x F y )  e.  C  ->  ( A F B )  e.  C ) )
104, 9mpi 17 1  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( A F B )  e.  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795    X. cxp 4938    Fn wfn 5513   -->wf 5514  (class class class)co 6192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fv 5526  df-ov 6195
This theorem is referenced by:  addclnq  9217  mulclnq  9219  adderpq  9228  mulerpq  9229  distrnq  9233  axaddcl  9421  axmulcl  9423  xaddcl  11310  xmulcl  11339  elfzoelz  11656  addcnlem  20558  sgmcl  22602  issubgoi  23934  ablomul  23979  hvaddcl  24551  hvmulcl  24552  hicl  24619  rmxynorm  29399  rmxyneg  29401  rmxy1  29403  rmxy0  29404  rmxp1  29413  rmyp1  29414  rmxm1  29415  rmym1  29416  rmxluc  29417  rmyluc  29418  rmyluc2  29419  rmxdbl  29420  rmydbl  29421  rmxypos  29430  ltrmynn0  29431  ltrmxnn0  29432  lermxnn0  29433  rmxnn  29434  ltrmy  29435  rmyeq0  29436  rmyeq  29437  lermy  29438  rmynn  29439  rmynn0  29440  rmyabs  29441  jm2.24nn  29442  jm2.17a  29443  jm2.17b  29444  jm2.17c  29445  jm2.24  29446  rmygeid  29447  jm2.18  29477  jm2.19lem1  29478  jm2.19lem2  29479  jm2.19  29482  jm2.22  29484  jm2.23  29485  jm2.20nn  29486  jm2.25  29488  jm2.26a  29489  jm2.26lem3  29490  jm2.26  29491  jm2.15nn0  29492  jm2.16nn0  29493  jm2.27a  29494  jm2.27c  29496  rmydioph  29503  rmxdiophlem  29504  jm3.1lem1  29506  jm3.1  29509  expdiophlem1  29510
  Copyright terms: Public domain W3C validator