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Theorem maprnin 26030
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 5558 . . . . . 6  |-  ( f : A --> B  -> 
f  Fn  A )
2 df-f 5421 . . . . . . 7  |-  ( f : A --> C  <->  ( f  Fn  A  /\  ran  f  C_  C ) )
32baibr 897 . . . . . 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 7240 . . . . 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 5590 . . . 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 2554 . 2  |-  { f  |  f : A --> ( B  i^i  C ) }  =  { f  |  ( f  e.  ( B  ^m  A
)  /\  ran  f  C_  C ) }
136inex1 4432 . . 3  |-  ( B  i^i  C )  e. 
_V
1413, 7mapval 7225 . 2  |-  ( ( B  i^i  C )  ^m  A )  =  { f  |  f : A --> ( B  i^i  C ) }
15 df-rab 2723 . 2  |-  { f  e.  ( B  ^m  A )  |  ran  f  C_  C }  =  { f  |  ( f  e.  ( B  ^m  A )  /\  ran  f  C_  C ) }
1612, 14, 153eqtr4i 2472 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 1369    e. wcel 1756   {cab 2428   {crab 2718   _Vcvv 2971    i^i cin 3326    C_ wss 3327   ran crn 4840    Fn wfn 5412   -->wf 5413  (class class class)co 6090    ^m cmap 7213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-map 7215
This theorem is referenced by:  fpwrelmapffs  26033
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