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Theorem maprnin 27254
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 5731 . . . . . 6  |-  ( f : A --> B  -> 
f  Fn  A )
2 df-f 5592 . . . . . . 7  |-  ( f : A --> C  <->  ( f  Fn  A  /\  ran  f  C_  C ) )
32baibr 902 . . . . . 6  |-  ( f  Fn  A  ->  ( ran  f  C_  C  <->  f : A
--> C ) )
41, 3syl 16 . . . . 5  |-  ( f : A --> B  -> 
( ran  f  C_  C 
<->  f : A --> C ) )
54pm5.32i 637 . . . 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 7447 . . . . 5  |-  ( f  e.  ( B  ^m  A )  <->  f : A
--> B )
98anbi1i 695 . . . 4  |-  ( ( f  e.  ( B  ^m  A )  /\  ran  f  C_  C )  <-> 
( f : A --> B  /\  ran  f  C_  C ) )
10 fin 5765 . . . 4  |-  ( f : A --> ( B  i^i  C )  <->  ( f : A --> B  /\  f : A --> C ) )
115, 9, 103bitr4ri 278 . . 3  |-  ( f : A --> ( B  i^i  C )  <->  ( f  e.  ( B  ^m  A
)  /\  ran  f  C_  C ) )
1211abbii 2601 . 2  |-  { f  |  f : A --> ( B  i^i  C ) }  =  { f  |  ( f  e.  ( B  ^m  A
)  /\  ran  f  C_  C ) }
136inex1 4588 . . 3  |-  ( B  i^i  C )  e. 
_V
1413, 7mapval 7432 . 2  |-  ( ( B  i^i  C )  ^m  A )  =  { f  |  f : A --> ( B  i^i  C ) }
15 df-rab 2823 . 2  |-  { f  e.  ( B  ^m  A )  |  ran  f  C_  C }  =  { f  |  ( f  e.  ( B  ^m  A )  /\  ran  f  C_  C ) }
1612, 14, 153eqtr4i 2506 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 184    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   {crab 2818   _Vcvv 3113    i^i cin 3475    C_ wss 3476   ran crn 5000    Fn wfn 5583   -->wf 5584  (class class class)co 6284    ^m cmap 7420
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-map 7422
This theorem is referenced by:  fpwrelmapffs  27257
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