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Theorem off 6109
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 3402 . . . . . . 7  |-  ( A  i^i  B )  C_  A
42, 3eqsstr3i 3222 . . . . . 6  |-  C  C_  A
54sseli 3189 . . . . 5  |-  ( z  e.  C  ->  z  e.  A )
6 ffvelrn 5679 . . . . 5  |-  ( ( F : A --> S  /\  z  e.  A )  ->  ( F `  z
)  e.  S )
71, 5, 6syl2an 463 . . . 4  |-  ( (
ph  /\  z  e.  C )  ->  ( F `  z )  e.  S )
8 off.3 . . . . 5  |-  ( ph  ->  G : B --> T )
9 inss2 3403 . . . . . . 7  |-  ( A  i^i  B )  C_  B
102, 9eqsstr3i 3222 . . . . . 6  |-  C  C_  B
1110sseli 3189 . . . . 5  |-  ( z  e.  C  ->  z  e.  B )
12 ffvelrn 5679 . . . . 5  |-  ( ( G : B --> T  /\  z  e.  B )  ->  ( G `  z
)  e.  T )
138, 11, 12syl2an 463 . . . 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 2648 . . . . 5  |-  ( ph  ->  A. x  e.  S  A. y  e.  T  ( x R y )  e.  U )
1615adantr 451 . . . 4  |-  ( (
ph  /\  z  e.  C )  ->  A. x  e.  S  A. y  e.  T  ( x R y )  e.  U )
17 oveq1 5881 . . . . . 6  |-  ( x  =  ( F `  z )  ->  (
x R y )  =  ( ( F `
 z ) R y ) )
1817eleq1d 2362 . . . . 5  |-  ( x  =  ( F `  z )  ->  (
( x R y )  e.  U  <->  ( ( F `  z ) R y )  e.  U ) )
19 oveq2 5882 . . . . . 6  |-  ( y  =  ( G `  z )  ->  (
( F `  z
) R y )  =  ( ( F `
 z ) R ( G `  z
) ) )
2019eleq1d 2362 . . . . 5  |-  ( y  =  ( G `  z )  ->  (
( ( F `  z ) R y )  e.  U  <->  ( ( F `  z ) R ( G `  z ) )  e.  U ) )
2118, 20rspc2va 2904 . . . 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 1181 . . 3  |-  ( (
ph  /\  z  e.  C )  ->  (
( F `  z
) R ( G `
 z ) )  e.  U )
23 eqid 2296 . . 3  |-  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) )  =  ( z  e.  C  |->  ( ( F `
 z ) R ( G `  z
) ) )
2422, 23fmptd 5700 . 2  |-  ( ph  ->  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) ) : C --> U )
25 ffn 5405 . . . . 5  |-  ( F : A --> S  ->  F  Fn  A )
261, 25syl 15 . . . 4  |-  ( ph  ->  F  Fn  A )
27 ffn 5405 . . . . 5  |-  ( G : B --> T  ->  G  Fn  B )
288, 27syl 15 . . . 4  |-  ( ph  ->  G  Fn  B )
29 off.4 . . . 4  |-  ( ph  ->  A  e.  V )
30 off.5 . . . 4  |-  ( ph  ->  B  e.  W )
31 eqidd 2297 . . . 4  |-  ( (
ph  /\  z  e.  A )  ->  ( F `  z )  =  ( F `  z ) )
32 eqidd 2297 . . . 4  |-  ( (
ph  /\  z  e.  B )  ->  ( G `  z )  =  ( G `  z ) )
3326, 28, 29, 30, 2, 31, 32offval 6101 . . 3  |-  ( ph  ->  ( F  o F R G )  =  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) ) )
3433feq1d 5395 . 2  |-  ( ph  ->  ( ( F  o F R G ) : C --> U  <->  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) ) : C --> U ) )
3524, 34mpbird 223 1  |-  ( ph  ->  ( F  o F R G ) : C --> U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    i^i cin 3164    e. cmpt 4093    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092
This theorem is referenced by:  o1of2  12102  ghmplusg  15154  gsumzaddlem  15219  gsumzadd  15220  psrbagaddcl  16132  psraddcl  16144  psrvscacl  16154  psrbagev1  16263  tsmsadd  17845  mbfmulc2lem  19018  mbfaddlem  19031  i1fadd  19066  i1fmul  19067  itg1addlem4  19070  i1fmulclem  19073  i1fmulc  19074  mbfi1flimlem  19093  itg2mulclem  19117  itg2mulc  19118  itg2monolem1  19121  itg2addlem  19129  dvaddbr  19303  dvmulbr  19304  dvaddf  19307  dvmulf  19308  dv11cn  19364  evlslem3  19414  plyaddlem  19613  coeeulem  19622  coeaddlem  19646  plydivlem4  19692  jensenlem2  20298  jensen  20299  basellem7  20340  basellem9  20342  dchrmulcl  20504  ofrn  23221  itg2addnclem  25003  itg2addnc  25005  lcomf  26870  mzpclall  26908  mzpindd  26927  frlmup1  27353  mndvcl  27549  expgrowth  27655  lfladdcl  29883  lflvscl  29889
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  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-of 6094
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