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Theorem maprnin 28322
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 7511 . . . . 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 2551 . 2  |-  { f  |  f : A --> ( B  i^i  C ) }  =  { f  |  ( f  e.  ( B  ^m  A
)  /\  ran  f  C_  C ) }
136inex1 4565 . . 3  |-  ( B  i^i  C )  e. 
_V
1413, 7mapval 7495 . 2  |-  ( ( B  i^i  C )  ^m  A )  =  { f  |  f : A --> ( B  i^i  C ) }
15 df-rab 2780 . 2  |-  { f  e.  ( B  ^m  A )  |  ran  f  C_  C }  =  { f  |  ( f  e.  ( B  ^m  A )  /\  ran  f  C_  C ) }
1612, 14, 153eqtr4i 2461 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 1872   {cab 2407   {crab 2775   _Vcvv 3080    i^i cin 3435    C_ wss 3436   ran crn 4854    Fn wfn 5596   -->wf 5597  (class class class)co 6305    ^m cmap 7483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  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 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  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 7485
This theorem is referenced by:  fpwrelmapffs  28325
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