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Theorem ofrn 26135
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 6307 . . . . . . . 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 2920 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  A. y  e.  B  ( x  .+  y )  e.  C
)
65r19.21bi 2920 . . . 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 3670 . . 3  |-  ( A  i^i  A )  =  A
127, 8, 9, 10, 10, 11off 6447 . 2  |-  ( ph  ->  ( F  oF  .+  G ) : A --> C )
13 frn 5676 . 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 1758   A.wral 2799    C_ wss 3439    X. cxp 4949   ran crn 4952    Fn wfn 5524   -->wf 5525  (class class class)co 6203    oFcof 6431
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pr 4642
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433
This theorem is referenced by: (None)
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