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Theorem ofrn2 24006
Description: The range of the function operation. (Contributed by Thierry Arnoux, 21-Mar-2017.)
Hypotheses
Ref Expression
ofrn.1  |-  ( ph  ->  F : A --> B )
ofrn.2  |-  ( ph  ->  G : A --> B )
ofrn.3  |-  ( ph  ->  .+  : ( B  X.  B ) --> C )
ofrn.4  |-  ( ph  ->  A  e.  V )
Assertion
Ref Expression
ofrn2  |-  ( ph  ->  ran  ( F  o F  .+  G )  C_  (  .+  " ( ran 
F  X.  ran  G
) ) )

Proof of Theorem ofrn2
Dummy variables  x  y  z  a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ofrn.1 . . . . . . . 8  |-  ( ph  ->  F : A --> B )
2 ffn 5550 . . . . . . . 8  |-  ( F : A --> B  ->  F  Fn  A )
31, 2syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  A )
43adantr 452 . . . . . 6  |-  ( (
ph  /\  ( a  e.  A  /\  z  =  ( ( F `
 a )  .+  ( G `  a ) ) ) )  ->  F  Fn  A )
5 simprl 733 . . . . . 6  |-  ( (
ph  /\  ( a  e.  A  /\  z  =  ( ( F `
 a )  .+  ( G `  a ) ) ) )  -> 
a  e.  A )
6 fnfvelrn 5826 . . . . . 6  |-  ( ( F  Fn  A  /\  a  e.  A )  ->  ( F `  a
)  e.  ran  F
)
74, 5, 6syl2anc 643 . . . . 5  |-  ( (
ph  /\  ( a  e.  A  /\  z  =  ( ( F `
 a )  .+  ( G `  a ) ) ) )  -> 
( F `  a
)  e.  ran  F
)
8 ofrn.2 . . . . . . . 8  |-  ( ph  ->  G : A --> B )
9 ffn 5550 . . . . . . . 8  |-  ( G : A --> B  ->  G  Fn  A )
108, 9syl 16 . . . . . . 7  |-  ( ph  ->  G  Fn  A )
1110adantr 452 . . . . . 6  |-  ( (
ph  /\  ( a  e.  A  /\  z  =  ( ( F `
 a )  .+  ( G `  a ) ) ) )  ->  G  Fn  A )
12 fnfvelrn 5826 . . . . . 6  |-  ( ( G  Fn  A  /\  a  e.  A )  ->  ( G `  a
)  e.  ran  G
)
1311, 5, 12syl2anc 643 . . . . 5  |-  ( (
ph  /\  ( a  e.  A  /\  z  =  ( ( F `
 a )  .+  ( G `  a ) ) ) )  -> 
( G `  a
)  e.  ran  G
)
14 simprr 734 . . . . 5  |-  ( (
ph  /\  ( a  e.  A  /\  z  =  ( ( F `
 a )  .+  ( G `  a ) ) ) )  -> 
z  =  ( ( F `  a ) 
.+  ( G `  a ) ) )
15 rspceov 6075 . . . . 5  |-  ( ( ( F `  a
)  e.  ran  F  /\  ( G `  a
)  e.  ran  G  /\  z  =  (
( F `  a
)  .+  ( G `  a ) ) )  ->  E. x  e.  ran  F E. y  e.  ran  G  z  =  ( x 
.+  y ) )
167, 13, 14, 15syl3anc 1184 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  z  =  ( ( F `
 a )  .+  ( G `  a ) ) ) )  ->  E. x  e.  ran  F E. y  e.  ran  G  z  =  ( x 
.+  y ) )
1716rexlimdvaa 2791 . . 3  |-  ( ph  ->  ( E. a  e.  A  z  =  ( ( F `  a
)  .+  ( G `  a ) )  ->  E. x  e.  ran  F E. y  e.  ran  G  z  =  ( x 
.+  y ) ) )
1817ss2abdv 3376 . 2  |-  ( ph  ->  { z  |  E. a  e.  A  z  =  ( ( F `
 a )  .+  ( G `  a ) ) }  C_  { z  |  E. x  e. 
ran  F E. y  e.  ran  G  z  =  ( x  .+  y
) } )
19 ofrn.4 . . . . 5  |-  ( ph  ->  A  e.  V )
20 inidm 3510 . . . . 5  |-  ( A  i^i  A )  =  A
21 eqidd 2405 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  ( F `  a )  =  ( F `  a ) )
22 eqidd 2405 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  ( G `  a )  =  ( G `  a ) )
233, 10, 19, 19, 20, 21, 22offval 6271 . . . 4  |-  ( ph  ->  ( F  o F 
.+  G )  =  ( a  e.  A  |->  ( ( F `  a )  .+  ( G `  a )
) ) )
2423rneqd 5056 . . 3  |-  ( ph  ->  ran  ( F  o F  .+  G )  =  ran  ( a  e.  A  |->  ( ( F `
 a )  .+  ( G `  a ) ) ) )
25 eqid 2404 . . . 4  |-  ( a  e.  A  |->  ( ( F `  a ) 
.+  ( G `  a ) ) )  =  ( a  e.  A  |->  ( ( F `
 a )  .+  ( G `  a ) ) )
2625rnmpt 5075 . . 3  |-  ran  (
a  e.  A  |->  ( ( F `  a
)  .+  ( G `  a ) ) )  =  { z  |  E. a  e.  A  z  =  ( ( F `  a )  .+  ( G `  a
) ) }
2724, 26syl6eq 2452 . 2  |-  ( ph  ->  ran  ( F  o F  .+  G )  =  { z  |  E. a  e.  A  z  =  ( ( F `
 a )  .+  ( G `  a ) ) } )
28 ofrn.3 . . . . 5  |-  ( ph  ->  .+  : ( B  X.  B ) --> C )
29 ffn 5550 . . . . 5  |-  (  .+  : ( B  X.  B ) --> C  ->  .+  Fn  ( B  X.  B ) )
3028, 29syl 16 . . . 4  |-  ( ph  ->  .+  Fn  ( B  X.  B ) )
31 frn 5556 . . . . . 6  |-  ( F : A --> B  ->  ran  F  C_  B )
321, 31syl 16 . . . . 5  |-  ( ph  ->  ran  F  C_  B
)
33 frn 5556 . . . . . 6  |-  ( G : A --> B  ->  ran  G  C_  B )
348, 33syl 16 . . . . 5  |-  ( ph  ->  ran  G  C_  B
)
35 xpss12 4940 . . . . 5  |-  ( ( ran  F  C_  B  /\  ran  G  C_  B
)  ->  ( ran  F  X.  ran  G ) 
C_  ( B  X.  B ) )
3632, 34, 35syl2anc 643 . . . 4  |-  ( ph  ->  ( ran  F  X.  ran  G )  C_  ( B  X.  B ) )
37 ovelimab 6183 . . . 4  |-  ( ( 
.+  Fn  ( B  X.  B )  /\  ( ran  F  X.  ran  G
)  C_  ( B  X.  B ) )  -> 
( z  e.  ( 
.+  " ( ran  F  X.  ran  G ) )  <->  E. x  e.  ran  F E. y  e.  ran  G  z  =  ( x 
.+  y ) ) )
3830, 36, 37syl2anc 643 . . 3  |-  ( ph  ->  ( z  e.  ( 
.+  " ( ran  F  X.  ran  G ) )  <->  E. x  e.  ran  F E. y  e.  ran  G  z  =  ( x 
.+  y ) ) )
3938abbi2dv 2519 . 2  |-  ( ph  ->  (  .+  " ( ran  F  X.  ran  G
) )  =  {
z  |  E. x  e.  ran  F E. y  e.  ran  G  z  =  ( x  .+  y
) } )
4018, 27, 393sstr4d 3351 1  |-  ( ph  ->  ran  ( F  o F  .+  G )  C_  (  .+  " ( ran 
F  X.  ran  G
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390   E.wrex 2667    C_ wss 3280    e. cmpt 4226    X. cxp 4835   ran crn 4838   "cima 4840    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262
This theorem is referenced by:  sibfof  24607
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264
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