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Theorem ofrn 27302
Description: The range of the function operation. (Contributed by Thierry Arnoux, 8-Jan-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
ofrn  |-  ( ph  ->  ran  ( F  oF  .+  G )  C_  C )

Proof of Theorem ofrn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ofrn.3 . . . . . . 7  |-  ( ph  ->  .+  : ( B  X.  B ) --> C )
2 ffnov 6401 . . . . . . . 8  |-  (  .+  : ( B  X.  B ) --> C  <->  (  .+  Fn  ( B  X.  B
)  /\  A. x  e.  B  A. y  e.  B  ( x  .+  y )  e.  C
) )
32simprbi 464 . . . . . . 7  |-  (  .+  : ( B  X.  B ) --> C  ->  A. x  e.  B  A. y  e.  B  ( x  .+  y )  e.  C )
41, 3syl 16 . . . . . 6  |-  ( ph  ->  A. x  e.  B  A. y  e.  B  ( x  .+  y )  e.  C )
54r19.21bi 2836 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  A. y  e.  B  ( x  .+  y )  e.  C
)
65r19.21bi 2836 . . . 4  |-  ( ( ( ph  /\  x  e.  B )  /\  y  e.  B )  ->  (
x  .+  y )  e.  C )
76anasss 647 . . 3  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x  .+  y
)  e.  C )
8 ofrn.1 . . 3  |-  ( ph  ->  F : A --> B )
9 ofrn.2 . . 3  |-  ( ph  ->  G : A --> B )
10 ofrn.4 . . 3  |-  ( ph  ->  A  e.  V )
11 inidm 3712 . . 3  |-  ( A  i^i  A )  =  A
127, 8, 9, 10, 10, 11off 6549 . 2  |-  ( ph  ->  ( F  oF  .+  G ) : A --> C )
13 frn 5743 . 2  |-  ( ( F  oF  .+  G ) : A --> C  ->  ran  ( F  oF  .+  G ) 
C_  C )
1412, 13syl 16 1  |-  ( ph  ->  ran  ( F  oF  .+  G )  C_  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767   A.wral 2817    C_ wss 3481    X. cxp 5003   ran crn 5006    Fn wfn 5589   -->wf 5590  (class class class)co 6295    oFcof 6533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535
This theorem is referenced by: (None)
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