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Theorem ofrn2 26094
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 5659 . . . . . . . 8  |-  ( F : A --> B  ->  F  Fn  A )
31, 2syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  A )
43adantr 465 . . . . . 6  |-  ( (
ph  /\  ( a  e.  A  /\  z  =  ( ( F `
 a )  .+  ( G `  a ) ) ) )  ->  F  Fn  A )
5 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( a  e.  A  /\  z  =  ( ( F `
 a )  .+  ( G `  a ) ) ) )  -> 
a  e.  A )
6 fnfvelrn 5941 . . . . . 6  |-  ( ( F  Fn  A  /\  a  e.  A )  ->  ( F `  a
)  e.  ran  F
)
74, 5, 6syl2anc 661 . . . . 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 5659 . . . . . . . 8  |-  ( G : A --> B  ->  G  Fn  A )
108, 9syl 16 . . . . . . 7  |-  ( ph  ->  G  Fn  A )
1110adantr 465 . . . . . 6  |-  ( (
ph  /\  ( a  e.  A  /\  z  =  ( ( F `
 a )  .+  ( G `  a ) ) ) )  ->  G  Fn  A )
12 fnfvelrn 5941 . . . . . 6  |-  ( ( G  Fn  A  /\  a  e.  A )  ->  ( G `  a
)  e.  ran  G
)
1311, 5, 12syl2anc 661 . . . . 5  |-  ( (
ph  /\  ( a  e.  A  /\  z  =  ( ( F `
 a )  .+  ( G `  a ) ) ) )  -> 
( G `  a
)  e.  ran  G
)
14 simprr 756 . . . . 5  |-  ( (
ph  /\  ( a  e.  A  /\  z  =  ( ( F `
 a )  .+  ( G `  a ) ) ) )  -> 
z  =  ( ( F `  a ) 
.+  ( G `  a ) ) )
15 rspceov 6229 . . . . 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 1219 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  z  =  ( ( F `
 a )  .+  ( G `  a ) ) ) )  ->  E. x  e.  ran  F E. y  e.  ran  G  z  =  ( x 
.+  y ) )
1716rexlimdvaa 2940 . . 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 3525 . 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 3659 . . . . 5  |-  ( A  i^i  A )  =  A
21 eqidd 2452 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  ( F `  a )  =  ( F `  a ) )
22 eqidd 2452 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  ( G `  a )  =  ( G `  a ) )
233, 10, 19, 19, 20, 21, 22offval 6429 . . . 4  |-  ( ph  ->  ( F  oF  .+  G )  =  ( a  e.  A  |->  ( ( F `  a )  .+  ( G `  a )
) ) )
2423rneqd 5167 . . 3  |-  ( ph  ->  ran  ( F  oF  .+  G )  =  ran  ( a  e.  A  |->  ( ( F `
 a )  .+  ( G `  a ) ) ) )
25 eqid 2451 . . . 4  |-  ( a  e.  A  |->  ( ( F `  a ) 
.+  ( G `  a ) ) )  =  ( a  e.  A  |->  ( ( F `
 a )  .+  ( G `  a ) ) )
2625rnmpt 5185 . . 3  |-  ran  (
a  e.  A  |->  ( ( F `  a
)  .+  ( G `  a ) ) )  =  { z  |  E. a  e.  A  z  =  ( ( F `  a )  .+  ( G `  a
) ) }
2724, 26syl6eq 2508 . 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 5659 . . . . 5  |-  (  .+  : ( B  X.  B ) --> C  ->  .+  Fn  ( B  X.  B ) )
3028, 29syl 16 . . . 4  |-  ( ph  ->  .+  Fn  ( B  X.  B ) )
31 frn 5665 . . . . . 6  |-  ( F : A --> B  ->  ran  F  C_  B )
321, 31syl 16 . . . . 5  |-  ( ph  ->  ran  F  C_  B
)
33 frn 5665 . . . . . 6  |-  ( G : A --> B  ->  ran  G  C_  B )
348, 33syl 16 . . . . 5  |-  ( ph  ->  ran  G  C_  B
)
35 xpss12 5045 . . . . 5  |-  ( ( ran  F  C_  B  /\  ran  G  C_  B
)  ->  ( ran  F  X.  ran  G ) 
C_  ( B  X.  B ) )
3632, 34, 35syl2anc 661 . . . 4  |-  ( ph  ->  ( ran  F  X.  ran  G )  C_  ( B  X.  B ) )
37 ovelimab 6343 . . . 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 661 . . 3  |-  ( ph  ->  ( z  e.  ( 
.+  " ( ran  F  X.  ran  G ) )  <->  E. x  e.  ran  F E. y  e.  ran  G  z  =  ( x 
.+  y ) ) )
3938abbi2dv 2588 . 2  |-  ( ph  ->  (  .+  " ( ran  F  X.  ran  G
) )  =  {
z  |  E. x  e.  ran  F E. y  e.  ran  G  z  =  ( x  .+  y
) } )
4018, 27, 393sstr4d 3499 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 369    = wceq 1370    e. wcel 1758   {cab 2436   E.wrex 2796    C_ wss 3428    |-> cmpt 4450    X. cxp 4938   ran crn 4941   "cima 4943    Fn wfn 5513   -->wf 5514   ` cfv 5518  (class class class)co 6192    oFcof 6420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-of 6422
This theorem is referenced by:  sibfof  26862
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