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Theorem off 6113
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 3370 . . . . . . 7  |-  ( A  i^i  B )  C_  A
42, 3eqsstrri 3203 . . . . . 6  |-  C  C_  A
54sseli 3166 . . . . 5  |-  ( z  e.  C  ->  z  e.  A )
6 ffvelcdm 5665 . . . . 5  |-  ( ( F : A --> S  /\  z  e.  A )  ->  ( F `  z
)  e.  S )
71, 5, 6syl2an 289 . . . 4  |-  ( (
ph  /\  z  e.  C )  ->  ( F `  z )  e.  S )
8 off.3 . . . . 5  |-  ( ph  ->  G : B --> T )
9 inss2 3371 . . . . . . 7  |-  ( A  i^i  B )  C_  B
102, 9eqsstrri 3203 . . . . . 6  |-  C  C_  B
1110sseli 3166 . . . . 5  |-  ( z  e.  C  ->  z  e.  B )
12 ffvelcdm 5665 . . . . 5  |-  ( ( G : B --> T  /\  z  e.  B )  ->  ( G `  z
)  e.  T )
138, 11, 12syl2an 289 . . . 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 2572 . . . . 5  |-  ( ph  ->  A. x  e.  S  A. y  e.  T  ( x R y )  e.  U )
1615adantr 276 . . . 4  |-  ( (
ph  /\  z  e.  C )  ->  A. x  e.  S  A. y  e.  T  ( x R y )  e.  U )
17 oveq1 5898 . . . . . 6  |-  ( x  =  ( F `  z )  ->  (
x R y )  =  ( ( F `
 z ) R y ) )
1817eleq1d 2258 . . . . 5  |-  ( x  =  ( F `  z )  ->  (
( x R y )  e.  U  <->  ( ( F `  z ) R y )  e.  U ) )
19 oveq2 5899 . . . . . 6  |-  ( y  =  ( G `  z )  ->  (
( F `  z
) R y )  =  ( ( F `
 z ) R ( G `  z
) ) )
2019eleq1d 2258 . . . . 5  |-  ( y  =  ( G `  z )  ->  (
( ( F `  z ) R y )  e.  U  <->  ( ( F `  z ) R ( G `  z ) )  e.  U ) )
2118, 20rspc2va 2870 . . . 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 1248 . . 3  |-  ( (
ph  /\  z  e.  C )  ->  (
( F `  z
) R ( G `
 z ) )  e.  U )
23 eqid 2189 . . 3  |-  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) )  =  ( z  e.  C  |->  ( ( F `
 z ) R ( G `  z
) ) )
2422, 23fmptd 5686 . 2  |-  ( ph  ->  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) ) : C --> U )
25 ffn 5380 . . . . 5  |-  ( F : A --> S  ->  F  Fn  A )
261, 25syl 14 . . . 4  |-  ( ph  ->  F  Fn  A )
27 ffn 5380 . . . . 5  |-  ( G : B --> T  ->  G  Fn  B )
288, 27syl 14 . . . 4  |-  ( ph  ->  G  Fn  B )
29 off.4 . . . 4  |-  ( ph  ->  A  e.  V )
30 off.5 . . . 4  |-  ( ph  ->  B  e.  W )
31 eqidd 2190 . . . 4  |-  ( (
ph  /\  z  e.  A )  ->  ( F `  z )  =  ( F `  z ) )
32 eqidd 2190 . . . 4  |-  ( (
ph  /\  z  e.  B )  ->  ( G `  z )  =  ( G `  z ) )
3326, 28, 29, 30, 2, 31, 32offval 6108 . . 3  |-  ( ph  ->  ( F  oF R G )  =  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) ) )
3433feq1d 5367 . 2  |-  ( ph  ->  ( ( F  oF R G ) : C --> U  <->  ( z  e.  C  |->  ( ( F `  z ) R ( G `  z ) ) ) : C --> U ) )
3524, 34mpbird 167 1  |-  ( ph  ->  ( F  oF R G ) : C --> U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   A.wral 2468    i^i cin 3143    |-> cmpt 4079    Fn wfn 5226   -->wf 5227   ` cfv 5231  (class class class)co 5891    oFcof 6099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-setind 4551
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-of 6101
This theorem is referenced by:  offeq  6114  lcomf  13604  dvaddxxbr  14549  dvmulxxbr  14550  dvaddxx  14551  dvmulxx  14552  dviaddf  14553  dvimulf  14554
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