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

Theorem off 6320
Description: The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.)
Hypotheses
Ref Expression
off.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  T ) )  -> 
( x R y )  e.  U )
off.2  |-  ( ph  ->  F : A --> S )
off.3  |-  ( ph  ->  G : B --> T )
off.4  |-  ( ph  ->  A  e.  V )
off.5  |-  ( ph  ->  B  e.  W )
off.6  |-  ( A  i^i  B )  =  C
Assertion
Ref Expression
off  |-  ( ph  ->  ( F  o F R G ) : C --> U )
Distinct variable groups:    y, G    x, y, ph    x, S, y   
x, T, y    x, F, y    x, R, y   
x, U, y
Allowed substitution hints:    A( x, y)    B( x, y)    C( x, y)    G( x)    V( x, y)    W( x, y)

Proof of Theorem off
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 off.2 . . . . 5  |-  ( ph  ->  F : A --> S )
2 off.6 . . . . . . 7  |-  ( A  i^i  B )  =  C
3 inss1 3561 . . . . . . 7  |-  ( A  i^i  B )  C_  A
42, 3eqsstr3i 3379 . . . . . 6  |-  C  C_  A
54sseli 3344 . . . . 5  |-  ( z  e.  C  ->  z  e.  A )
6 ffvelrn 5868 . . . . 5  |-  ( ( F : A --> S  /\  z  e.  A )  ->  ( F `  z
)  e.  S )
71, 5, 6syl2an 464 . . . 4  |-  ( (
ph  /\  z  e.  C )  ->  ( F `  z )  e.  S )
8 off.3 . . . . 5  |-  ( ph  ->  G : B --> T )
9 inss2 3562 . . . . . . 7  |-  ( A  i^i  B )  C_  B
102, 9eqsstr3i 3379 . . . . . 6  |-  C  C_  B
1110sseli 3344 . . . . 5  |-  ( z  e.  C  ->  z  e.  B )
12 ffvelrn 5868 . . . . 5  |-  ( ( G : B --> T  /\  z  e.  B )  ->  ( G `  z
)  e.  T )
138, 11, 12syl2an 464 . . . 4  |-  ( (
ph  /\  z  e.  C )  ->  ( G `  z )  e.  T )
14 off.1 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  T ) )  -> 
( x R y )  e.  U )
1514ralrimivva 2798 . . . . 5  |-  ( ph  ->  A. x  e.  S  A. y  e.  T  ( x R y )  e.  U )
1615adantr 452 . . . 4  |-  ( (
ph  /\  z  e.  C )  ->  A. x  e.  S  A. y  e.  T  ( x R y )  e.  U )
17 oveq1 6088 . . . . . 6  |-  ( x  =  ( F `  z )  ->  (
x R y )  =  ( ( F `
 z ) R y ) )
1817eleq1d 2502 . . . . 5  |-  ( x  =  ( F `  z )  ->  (
( x R y )  e.  U  <->  ( ( F `  z ) R y )  e.  U ) )
19 oveq2 6089 . . . . . 6  |-  ( y  =  ( G `  z )  ->  (
( F `  z
) R y )  =  ( ( F `
 z ) R ( G `  z
) ) )
2019eleq1d 2502 . . . . 5  |-  ( y  =  ( G `  z )  ->  (
( ( F `  z ) R y )  e.  U  <->  ( ( F `  z ) R ( G `  z ) )  e.  U ) )
2118, 20rspc2va 3059 . . . 4  |-  ( ( ( ( F `  z )  e.  S  /\  ( G `  z
)  e.  T )  /\  A. x  e.  S  A. y  e.  T  ( x R y )  e.  U
)  ->  ( ( F `  z ) R ( G `  z ) )  e.  U )
227, 13, 16, 21syl21anc 1183 . . 3  |-  ( (
ph  /\  z  e.  C )  ->  (
( F `  z
) R ( G `
 z ) )  e.  U )
23 eqid 2436 . . 3  |-  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) )  =  ( z  e.  C  |->  ( ( F `
 z ) R ( G `  z
) ) )
2422, 23fmptd 5893 . 2  |-  ( ph  ->  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) ) : C --> U )
25 ffn 5591 . . . . 5  |-  ( F : A --> S  ->  F  Fn  A )
261, 25syl 16 . . . 4  |-  ( ph  ->  F  Fn  A )
27 ffn 5591 . . . . 5  |-  ( G : B --> T  ->  G  Fn  B )
288, 27syl 16 . . . 4  |-  ( ph  ->  G  Fn  B )
29 off.4 . . . 4  |-  ( ph  ->  A  e.  V )
30 off.5 . . . 4  |-  ( ph  ->  B  e.  W )
31 eqidd 2437 . . . 4  |-  ( (
ph  /\  z  e.  A )  ->  ( F `  z )  =  ( F `  z ) )
32 eqidd 2437 . . . 4  |-  ( (
ph  /\  z  e.  B )  ->  ( G `  z )  =  ( G `  z ) )
3326, 28, 29, 30, 2, 31, 32offval 6312 . . 3  |-  ( ph  ->  ( F  o F R G )  =  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) ) )
3433feq1d 5580 . 2  |-  ( ph  ->  ( ( F  o F R G ) : C --> U  <->  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) ) : C --> U ) )
3524, 34mpbird 224 1  |-  ( ph  ->  ( F  o F R G ) : C --> U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705    i^i cin 3319    e. cmpt 4266    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303
This theorem is referenced by:  o1of2  12406  ghmplusg  15461  gsumzaddlem  15526  gsumzadd  15527  psrbagaddcl  16435  psraddcl  16447  psrvscacl  16457  psrbagev1  16566  tsmsadd  18176  mbfmulc2lem  19539  mbfaddlem  19552  i1fadd  19587  i1fmul  19588  itg1addlem4  19591  i1fmulclem  19594  i1fmulc  19595  mbfi1flimlem  19614  itg2mulclem  19638  itg2mulc  19639  itg2monolem1  19642  itg2addlem  19650  dvaddbr  19824  dvmulbr  19825  dvaddf  19828  dvmulf  19829  dv11cn  19885  evlslem3  19935  plyaddlem  20134  coeeulem  20143  coeaddlem  20167  plydivlem4  20213  jensenlem2  20826  jensen  20827  basellem7  20869  basellem9  20871  dchrmulcl  21033  ofrn  24052  sibfof  24654  itg2addnc  26259  ftc1anclem3  26282  ftc1anclem6  26285  ftc1anclem8  26287  lcomf  26746  mzpclall  26784  mzpindd  26803  frlmup1  27227  mndvcl  27423  expgrowth  27529  lfladdcl  29869  lflvscl  29875
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305
  Copyright terms: Public domain W3C validator