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Theorem off 6560
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  oF 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 3688 . . . . . . 7  |-  ( A  i^i  B )  C_  A
42, 3eqsstr3i 3501 . . . . . 6  |-  C  C_  A
54sseli 3466 . . . . 5  |-  ( z  e.  C  ->  z  e.  A )
6 ffvelrn 6035 . . . . 5  |-  ( ( F : A --> S  /\  z  e.  A )  ->  ( F `  z
)  e.  S )
71, 5, 6syl2an 479 . . . 4  |-  ( (
ph  /\  z  e.  C )  ->  ( F `  z )  e.  S )
8 off.3 . . . . 5  |-  ( ph  ->  G : B --> T )
9 inss2 3689 . . . . . . 7  |-  ( A  i^i  B )  C_  B
102, 9eqsstr3i 3501 . . . . . 6  |-  C  C_  B
1110sseli 3466 . . . . 5  |-  ( z  e.  C  ->  z  e.  B )
12 ffvelrn 6035 . . . . 5  |-  ( ( G : B --> T  /\  z  e.  B )  ->  ( G `  z
)  e.  T )
138, 11, 12syl2an 479 . . . 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 2853 . . . . 5  |-  ( ph  ->  A. x  e.  S  A. y  e.  T  ( x R y )  e.  U )
1615adantr 466 . . . 4  |-  ( (
ph  /\  z  e.  C )  ->  A. x  e.  S  A. y  e.  T  ( x R y )  e.  U )
17 oveq1 6312 . . . . . 6  |-  ( x  =  ( F `  z )  ->  (
x R y )  =  ( ( F `
 z ) R y ) )
1817eleq1d 2498 . . . . 5  |-  ( x  =  ( F `  z )  ->  (
( x R y )  e.  U  <->  ( ( F `  z ) R y )  e.  U ) )
19 oveq2 6313 . . . . . 6  |-  ( y  =  ( G `  z )  ->  (
( F `  z
) R y )  =  ( ( F `
 z ) R ( G `  z
) ) )
2019eleq1d 2498 . . . . 5  |-  ( y  =  ( G `  z )  ->  (
( ( F `  z ) R y )  e.  U  <->  ( ( F `  z ) R ( G `  z ) )  e.  U ) )
2118, 20rspc2va 3198 . . . 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 1263 . . 3  |-  ( (
ph  /\  z  e.  C )  ->  (
( F `  z
) R ( G `
 z ) )  e.  U )
23 eqid 2429 . . 3  |-  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) )  =  ( z  e.  C  |->  ( ( F `
 z ) R ( G `  z
) ) )
2422, 23fmptd 6061 . 2  |-  ( ph  ->  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) ) : C --> U )
25 ffn 5746 . . . . 5  |-  ( F : A --> S  ->  F  Fn  A )
261, 25syl 17 . . . 4  |-  ( ph  ->  F  Fn  A )
27 ffn 5746 . . . . 5  |-  ( G : B --> T  ->  G  Fn  B )
288, 27syl 17 . . . 4  |-  ( ph  ->  G  Fn  B )
29 off.4 . . . 4  |-  ( ph  ->  A  e.  V )
30 off.5 . . . 4  |-  ( ph  ->  B  e.  W )
31 eqidd 2430 . . . 4  |-  ( (
ph  /\  z  e.  A )  ->  ( F `  z )  =  ( F `  z ) )
32 eqidd 2430 . . . 4  |-  ( (
ph  /\  z  e.  B )  ->  ( G `  z )  =  ( G `  z ) )
3326, 28, 29, 30, 2, 31, 32offval 6552 . . 3  |-  ( ph  ->  ( F  oF R G )  =  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) ) )
3433feq1d 5732 . 2  |-  ( ph  ->  ( ( F  oF R G ) : C --> U  <->  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) ) : C --> U ) )
3524, 34mpbird 235 1  |-  ( ph  ->  ( F  oF R G ) : C --> U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782    i^i cin 3441    |-> cmpt 4484    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305    oFcof 6543
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 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545
This theorem is referenced by:  o1of2  13654  ghmplusg  17419  gsumzaddlem  17489  gsumzadd  17490  lcomf  18062  psrbagaddcl  18529  psraddcl  18542  psrvscacl  18552  psrbagev1  18668  evlslem3  18672  frlmup1  19287  mndvcl  19347  tsmsadd  21092  mbfmulc2lem  22480  mbfaddlem  22493  i1fadd  22530  i1fmul  22531  itg1addlem4  22534  i1fmulclem  22537  i1fmulc  22538  mbfi1flimlem  22557  itg2mulclem  22581  itg2mulc  22582  itg2monolem1  22585  itg2addlem  22593  dvaddbr  22769  dvmulbr  22770  dvaddf  22773  dvmulf  22774  dv11cn  22830  plyaddlem  23037  coeeulem  23046  coeaddlem  23071  plydivlem4  23117  jensenlem2  23778  jensen  23779  basellem7  23876  basellem9  23878  dchrmulcl  24040  ofrn  28080  sibfof  29001  signshf  29265  poimirlem23  31667  poimirlem24  31668  poimirlem25  31669  poimirlem29  31673  poimirlem30  31674  poimirlem31  31675  poimirlem32  31676  itg2addnc  31700  ftc1anclem3  31723  ftc1anclem6  31726  ftc1anclem8  31728  lfladdcl  32346  lflvscl  32352  mzpclall  35278  mzpindd  35297  expgrowth  36321  binomcxplemnotnn0  36342  dvdivcncf  37371  ofaddmndmap  38894
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