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Theorem ofrn2 27003
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 5722 . . . . . . . 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 6009 . . . . . 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 5722 . . . . . . . 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 6009 . . . . . 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 6312 . . . . 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 1223 . . . 4  |-  ( (
ph  /\  ( a  e.  A  /\  z  =  ( ( F `
 a )  .+  ( G `  a ) ) ) )  ->  E. x  e.  ran  F E. y  e.  ran  G  z  =  ( x 
.+  y ) )
1716rexlimdvaa 2949 . . 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 3566 . 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 3700 . . . . 5  |-  ( A  i^i  A )  =  A
21 eqidd 2461 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  ( F `  a )  =  ( F `  a ) )
22 eqidd 2461 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  ( G `  a )  =  ( G `  a ) )
233, 10, 19, 19, 20, 21, 22offval 6522 . . . 4  |-  ( ph  ->  ( F  oF  .+  G )  =  ( a  e.  A  |->  ( ( F `  a )  .+  ( G `  a )
) ) )
2423rneqd 5221 . . 3  |-  ( ph  ->  ran  ( F  oF  .+  G )  =  ran  ( a  e.  A  |->  ( ( F `
 a )  .+  ( G `  a ) ) ) )
25 eqid 2460 . . . 4  |-  ( a  e.  A  |->  ( ( F `  a ) 
.+  ( G `  a ) ) )  =  ( a  e.  A  |->  ( ( F `
 a )  .+  ( G `  a ) ) )
2625rnmpt 5239 . . 3  |-  ran  (
a  e.  A  |->  ( ( F `  a
)  .+  ( G `  a ) ) )  =  { z  |  E. a  e.  A  z  =  ( ( F `  a )  .+  ( G `  a
) ) }
2724, 26syl6eq 2517 . 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 5722 . . . . 5  |-  (  .+  : ( B  X.  B ) --> C  ->  .+  Fn  ( B  X.  B ) )
3028, 29syl 16 . . . 4  |-  ( ph  ->  .+  Fn  ( B  X.  B ) )
31 frn 5728 . . . . . 6  |-  ( F : A --> B  ->  ran  F  C_  B )
321, 31syl 16 . . . . 5  |-  ( ph  ->  ran  F  C_  B
)
33 frn 5728 . . . . . 6  |-  ( G : A --> B  ->  ran  G  C_  B )
348, 33syl 16 . . . . 5  |-  ( ph  ->  ran  G  C_  B
)
35 xpss12 5099 . . . . 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 6428 . . . 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 2597 . 2  |-  ( ph  ->  (  .+  " ( ran  F  X.  ran  G
) )  =  {
z  |  E. x  e.  ran  F E. y  e.  ran  G  z  =  ( x  .+  y
) } )
4018, 27, 393sstr4d 3540 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 1374    e. wcel 1762   {cab 2445   E.wrex 2808    C_ wss 3469    |-> cmpt 4498    X. cxp 4990   ran crn 4993   "cima 4995    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275    oFcof 6513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515
This theorem is referenced by:  sibfof  27772
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