Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  maprnin Structured version   Unicode version

Theorem maprnin 28159
Description: Restricting the range of the mapping operator (Contributed by Thierry Arnoux, 30-Aug-2017.)
Hypotheses
Ref Expression
maprnin.1  |-  A  e. 
_V
maprnin.2  |-  B  e. 
_V
Assertion
Ref Expression
maprnin  |-  ( ( B  i^i  C )  ^m  A )  =  { f  e.  ( B  ^m  A )  |  ran  f  C_  C }
Distinct variable groups:    A, f    B, f    C, f

Proof of Theorem maprnin
StepHypRef Expression
1 ffn 5746 . . . . . 6  |-  ( f : A --> B  -> 
f  Fn  A )
2 df-f 5605 . . . . . . 7  |-  ( f : A --> C  <->  ( f  Fn  A  /\  ran  f  C_  C ) )
32baibr 912 . . . . . 6  |-  ( f  Fn  A  ->  ( ran  f  C_  C  <->  f : A
--> C ) )
41, 3syl 17 . . . . 5  |-  ( f : A --> B  -> 
( ran  f  C_  C 
<->  f : A --> C ) )
54pm5.32i 641 . . . 4  |-  ( ( f : A --> B  /\  ran  f  C_  C )  <-> 
( f : A --> B  /\  f : A --> C ) )
6 maprnin.2 . . . . . 6  |-  B  e. 
_V
7 maprnin.1 . . . . . 6  |-  A  e. 
_V
86, 7elmap 7508 . . . . 5  |-  ( f  e.  ( B  ^m  A )  <->  f : A
--> B )
98anbi1i 699 . . . 4  |-  ( ( f  e.  ( B  ^m  A )  /\  ran  f  C_  C )  <-> 
( f : A --> B  /\  ran  f  C_  C ) )
10 fin 5780 . . . 4  |-  ( f : A --> ( B  i^i  C )  <->  ( f : A --> B  /\  f : A --> C ) )
115, 9, 103bitr4ri 281 . . 3  |-  ( f : A --> ( B  i^i  C )  <->  ( f  e.  ( B  ^m  A
)  /\  ran  f  C_  C ) )
1211abbii 2563 . 2  |-  { f  |  f : A --> ( B  i^i  C ) }  =  { f  |  ( f  e.  ( B  ^m  A
)  /\  ran  f  C_  C ) }
136inex1 4566 . . 3  |-  ( B  i^i  C )  e. 
_V
1413, 7mapval 7492 . 2  |-  ( ( B  i^i  C )  ^m  A )  =  { f  |  f : A --> ( B  i^i  C ) }
15 df-rab 2791 . 2  |-  { f  e.  ( B  ^m  A )  |  ran  f  C_  C }  =  { f  |  ( f  e.  ( B  ^m  A )  /\  ran  f  C_  C ) }
1612, 14, 153eqtr4i 2468 1  |-  ( ( B  i^i  C )  ^m  A )  =  { f  e.  ( B  ^m  A )  |  ran  f  C_  C }
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   {cab 2414   {crab 2786   _Vcvv 3087    i^i cin 3441    C_ wss 3442   ran crn 4855    Fn wfn 5596   -->wf 5597  (class class class)co 6305    ^m cmap 7480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-map 7482
This theorem is referenced by:  fpwrelmapffs  28162
  Copyright terms: Public domain W3C validator