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Theorem ofrn2 27923
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  oF  .+  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 5714 . . . . . . . 8  |-  ( F : A --> B  ->  F  Fn  A )
31, 2syl 17 . . . . . . 7  |-  ( ph  ->  F  Fn  A )
43adantr 463 . . . . . 6  |-  ( (
ph  /\  ( a  e.  A  /\  z  =  ( ( F `
 a )  .+  ( G `  a ) ) ) )  ->  F  Fn  A )
5 simprl 756 . . . . . 6  |-  ( (
ph  /\  ( a  e.  A  /\  z  =  ( ( F `
 a )  .+  ( G `  a ) ) ) )  -> 
a  e.  A )
6 fnfvelrn 6006 . . . . . 6  |-  ( ( F  Fn  A  /\  a  e.  A )  ->  ( F `  a
)  e.  ran  F
)
74, 5, 6syl2anc 659 . . . . 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 5714 . . . . . . . 8  |-  ( G : A --> B  ->  G  Fn  A )
108, 9syl 17 . . . . . . 7  |-  ( ph  ->  G  Fn  A )
1110adantr 463 . . . . . 6  |-  ( (
ph  /\  ( a  e.  A  /\  z  =  ( ( F `
 a )  .+  ( G `  a ) ) ) )  ->  G  Fn  A )
12 fnfvelrn 6006 . . . . . 6  |-  ( ( G  Fn  A  /\  a  e.  A )  ->  ( G `  a
)  e.  ran  G
)
1311, 5, 12syl2anc 659 . . . . 5  |-  ( (
ph  /\  ( a  e.  A  /\  z  =  ( ( F `
 a )  .+  ( G `  a ) ) ) )  -> 
( G `  a
)  e.  ran  G
)
14 simprr 758 . . . . 5  |-  ( (
ph  /\  ( a  e.  A  /\  z  =  ( ( F `
 a )  .+  ( G `  a ) ) ) )  -> 
z  =  ( ( F `  a ) 
.+  ( G `  a ) ) )
15 rspceov 6317 . . . . 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 1230 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  z  =  ( ( F `
 a )  .+  ( G `  a ) ) ) )  ->  E. x  e.  ran  F E. y  e.  ran  G  z  =  ( x 
.+  y ) )
1716rexlimdvaa 2897 . . 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 3512 . 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 3648 . . . . 5  |-  ( A  i^i  A )  =  A
21 eqidd 2403 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  ( F `  a )  =  ( F `  a ) )
22 eqidd 2403 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  ( G `  a )  =  ( G `  a ) )
233, 10, 19, 19, 20, 21, 22offval 6528 . . . 4  |-  ( ph  ->  ( F  oF  .+  G )  =  ( a  e.  A  |->  ( ( F `  a )  .+  ( G `  a )
) ) )
2423rneqd 5051 . . 3  |-  ( ph  ->  ran  ( F  oF  .+  G )  =  ran  ( a  e.  A  |->  ( ( F `
 a )  .+  ( G `  a ) ) ) )
25 eqid 2402 . . . 4  |-  ( a  e.  A  |->  ( ( F `  a ) 
.+  ( G `  a ) ) )  =  ( a  e.  A  |->  ( ( F `
 a )  .+  ( G `  a ) ) )
2625rnmpt 5069 . . 3  |-  ran  (
a  e.  A  |->  ( ( F `  a
)  .+  ( G `  a ) ) )  =  { z  |  E. a  e.  A  z  =  ( ( F `  a )  .+  ( G `  a
) ) }
2724, 26syl6eq 2459 . 2  |-  ( ph  ->  ran  ( F  oF  .+  G )  =  { z  |  E. a  e.  A  z  =  ( ( F `
 a )  .+  ( G `  a ) ) } )
28 ofrn.3 . . . . 5  |-  ( ph  ->  .+  : ( B  X.  B ) --> C )
29 ffn 5714 . . . . 5  |-  (  .+  : ( B  X.  B ) --> C  ->  .+  Fn  ( B  X.  B ) )
3028, 29syl 17 . . . 4  |-  ( ph  ->  .+  Fn  ( B  X.  B ) )
31 frn 5720 . . . . . 6  |-  ( F : A --> B  ->  ran  F  C_  B )
321, 31syl 17 . . . . 5  |-  ( ph  ->  ran  F  C_  B
)
33 frn 5720 . . . . . 6  |-  ( G : A --> B  ->  ran  G  C_  B )
348, 33syl 17 . . . . 5  |-  ( ph  ->  ran  G  C_  B
)
35 xpss12 4929 . . . . 5  |-  ( ( ran  F  C_  B  /\  ran  G  C_  B
)  ->  ( ran  F  X.  ran  G ) 
C_  ( B  X.  B ) )
3632, 34, 35syl2anc 659 . . . 4  |-  ( ph  ->  ( ran  F  X.  ran  G )  C_  ( B  X.  B ) )
37 ovelimab 6434 . . . 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 659 . . 3  |-  ( ph  ->  ( z  e.  ( 
.+  " ( ran  F  X.  ran  G ) )  <->  E. x  e.  ran  F E. y  e.  ran  G  z  =  ( x 
.+  y ) ) )
3938abbi2dv 2539 . 2  |-  ( ph  ->  (  .+  " ( ran  F  X.  ran  G
) )  =  {
z  |  E. x  e.  ran  F E. y  e.  ran  G  z  =  ( x  .+  y
) } )
4018, 27, 393sstr4d 3485 1  |-  ( ph  ->  ran  ( F  oF  .+  G )  C_  (  .+  " ( ran 
F  X.  ran  G
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   {cab 2387   E.wrex 2755    C_ wss 3414    |-> cmpt 4453    X. cxp 4821   ran crn 4824   "cima 4826    Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278    oFcof 6519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521
This theorem is referenced by:  sibfof  28788
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