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Theorem off 6538
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 3718 . . . . . . 7  |-  ( A  i^i  B )  C_  A
42, 3eqsstr3i 3535 . . . . . 6  |-  C  C_  A
54sseli 3500 . . . . 5  |-  ( z  e.  C  ->  z  e.  A )
6 ffvelrn 6019 . . . . 5  |-  ( ( F : A --> S  /\  z  e.  A )  ->  ( F `  z
)  e.  S )
71, 5, 6syl2an 477 . . . 4  |-  ( (
ph  /\  z  e.  C )  ->  ( F `  z )  e.  S )
8 off.3 . . . . 5  |-  ( ph  ->  G : B --> T )
9 inss2 3719 . . . . . . 7  |-  ( A  i^i  B )  C_  B
102, 9eqsstr3i 3535 . . . . . 6  |-  C  C_  B
1110sseli 3500 . . . . 5  |-  ( z  e.  C  ->  z  e.  B )
12 ffvelrn 6019 . . . . 5  |-  ( ( G : B --> T  /\  z  e.  B )  ->  ( G `  z
)  e.  T )
138, 11, 12syl2an 477 . . . 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 2885 . . . . 5  |-  ( ph  ->  A. x  e.  S  A. y  e.  T  ( x R y )  e.  U )
1615adantr 465 . . . 4  |-  ( (
ph  /\  z  e.  C )  ->  A. x  e.  S  A. y  e.  T  ( x R y )  e.  U )
17 oveq1 6291 . . . . . 6  |-  ( x  =  ( F `  z )  ->  (
x R y )  =  ( ( F `
 z ) R y ) )
1817eleq1d 2536 . . . . 5  |-  ( x  =  ( F `  z )  ->  (
( x R y )  e.  U  <->  ( ( F `  z ) R y )  e.  U ) )
19 oveq2 6292 . . . . . 6  |-  ( y  =  ( G `  z )  ->  (
( F `  z
) R y )  =  ( ( F `
 z ) R ( G `  z
) ) )
2019eleq1d 2536 . . . . 5  |-  ( y  =  ( G `  z )  ->  (
( ( F `  z ) R y )  e.  U  <->  ( ( F `  z ) R ( G `  z ) )  e.  U ) )
2118, 20rspc2va 3224 . . . 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 1227 . . 3  |-  ( (
ph  /\  z  e.  C )  ->  (
( F `  z
) R ( G `
 z ) )  e.  U )
23 eqid 2467 . . 3  |-  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) )  =  ( z  e.  C  |->  ( ( F `
 z ) R ( G `  z
) ) )
2422, 23fmptd 6045 . 2  |-  ( ph  ->  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) ) : C --> U )
25 ffn 5731 . . . . 5  |-  ( F : A --> S  ->  F  Fn  A )
261, 25syl 16 . . . 4  |-  ( ph  ->  F  Fn  A )
27 ffn 5731 . . . . 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 2468 . . . 4  |-  ( (
ph  /\  z  e.  A )  ->  ( F `  z )  =  ( F `  z ) )
32 eqidd 2468 . . . 4  |-  ( (
ph  /\  z  e.  B )  ->  ( G `  z )  =  ( G `  z ) )
3326, 28, 29, 30, 2, 31, 32offval 6531 . . 3  |-  ( ph  ->  ( F  oF R G )  =  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) ) )
3433feq1d 5717 . 2  |-  ( ph  ->  ( ( F  oF R G ) : C --> U  <->  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) ) : C --> U ) )
3524, 34mpbird 232 1  |-  ( ph  ->  ( F  oF R G ) : C --> U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814    i^i cin 3475    |-> cmpt 4505    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284    oFcof 6522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524
This theorem is referenced by:  o1of2  13398  ghmplusg  16655  gsumzaddlem  16737  gsumzadd  16738  gsumzaddlemOLD  16739  gsumzaddOLD  16740  lcomf  17348  psrbagaddcl  17819  psrbagaddclOLD  17820  psraddcl  17835  psrvscacl  17845  psrbagev1  17976  psrbagev1OLD  17977  evlslem3  17982  frlmup1  18627  mndvcl  18688  tsmsadd  20412  mbfmulc2lem  21817  mbfaddlem  21830  i1fadd  21865  i1fmul  21866  itg1addlem4  21869  i1fmulclem  21872  i1fmulc  21873  mbfi1flimlem  21892  itg2mulclem  21916  itg2mulc  21917  itg2monolem1  21920  itg2addlem  21928  dvaddbr  22104  dvmulbr  22105  dvaddf  22108  dvmulf  22109  dv11cn  22165  plyaddlem  22375  coeeulem  22384  coeaddlem  22408  plydivlem4  22454  jensenlem2  23073  jensen  23074  basellem7  23116  basellem9  23118  dchrmulcl  23280  ofrn  27180  sibfof  27950  signshf  28213  itg2addnc  29674  ftc1anclem3  29697  ftc1anclem6  29700  ftc1anclem8  29702  mzpclall  30291  mzpindd  30310  expgrowth  30868  dvdivcncf  31285  ofaddmndmap  32029  lfladdcl  33886  lflvscl  33892
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