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Theorem mapex2 25140
Description: Two ways to express a subset of mappings. (Contributed by FL, 17-Nov-2014.)
Assertion
Ref Expression
mapex2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  ( f : A --> B  /\  ph ) }  =  {
f  e.  ( B  ^m  A )  | 
ph } )
Distinct variable groups:    A, f    B, f
Allowed substitution hints:    ph( f)    C( f)    D( f)

Proof of Theorem mapex2
StepHypRef Expression
1 inab 3436 . 2  |-  ( { f  |  f : A --> B }  i^i  { f  |  ph }
)  =  { f  |  ( f : A --> B  /\  ph ) }
2 mapvalg 6782 . . . . . 6  |-  ( ( B  e.  D  /\  A  e.  C )  ->  ( B  ^m  A
)  =  { f  |  f : A --> B } )
32ancoms 439 . . . . 5  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( B  ^m  A
)  =  { f  |  f : A --> B } )
43eqcomd 2288 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  f : A --> B }  =  ( B  ^m  A ) )
54ineq1d 3369 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { f  |  f : A --> B }  i^i  { f  |  ph } )  =  ( ( B  ^m  A
)  i^i  { f  |  ph } ) )
6 abid2 2400 . . . . . 6  |-  { f  |  f  e.  ( B  ^m  A ) }  =  ( B  ^m  A )
76eqcomi 2287 . . . . 5  |-  ( B  ^m  A )  =  { f  |  f  e.  ( B  ^m  A ) }
87a1i 10 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( B  ^m  A
)  =  { f  |  f  e.  ( B  ^m  A ) } )
98ineq1d 3369 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( B  ^m  A )  i^i  {
f  |  ph }
)  =  ( { f  |  f  e.  ( B  ^m  A
) }  i^i  {
f  |  ph }
) )
10 inab 3436 . . . . 5  |-  ( { f  |  f  e.  ( B  ^m  A
) }  i^i  {
f  |  ph }
)  =  { f  |  ( f  e.  ( B  ^m  A
)  /\  ph ) }
1110a1i 10 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { f  |  f  e.  ( B  ^m  A ) }  i^i  { f  | 
ph } )  =  { f  |  ( f  e.  ( B  ^m  A )  /\  ph ) } )
12 df-rab 2552 . . . 4  |-  { f  e.  ( B  ^m  A )  |  ph }  =  { f  |  ( f  e.  ( B  ^m  A
)  /\  ph ) }
1311, 12syl6eqr 2333 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { f  |  f  e.  ( B  ^m  A ) }  i^i  { f  | 
ph } )  =  { f  e.  ( B  ^m  A )  |  ph } )
145, 9, 133eqtrd 2319 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { f  |  f : A --> B }  i^i  { f  |  ph } )  =  {
f  e.  ( B  ^m  A )  | 
ph } )
151, 14syl5eqr 2329 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  ( f : A --> B  /\  ph ) }  =  {
f  e.  ( B  ^m  A )  | 
ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   {crab 2547    i^i cin 3151   -->wf 5251  (class class class)co 5858    ^m cmap 6772
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774
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