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Theorem ofcfval2 23663
Description: The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 31-Jan-2017.)
Hypotheses
Ref Expression
ofcfval2.1  |-  ( ph  ->  A  e.  V )
ofcfval2.2  |-  ( ph  ->  C  e.  W )
ofcfval2.3  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  W )
ofcfval2.4  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
Assertion
Ref Expression
ofcfval2  |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( x  e.  A  |->  ( B R C ) ) )
Distinct variable groups:    x, A    x, C    x, F    x, R    ph, x
Allowed substitution hints:    B( x)    V( x)    W( x)

Proof of Theorem ofcfval2
StepHypRef Expression
1 ofcfval2.3 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  W )
21ralrimiva 2660 . . . 4  |-  ( ph  ->  A. x  e.  A  B  e.  W )
3 eqid 2316 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
43fnmpt 5407 . . . 4  |-  ( A. x  e.  A  B  e.  W  ->  ( x  e.  A  |->  B )  Fn  A )
52, 4syl 15 . . 3  |-  ( ph  ->  ( x  e.  A  |->  B )  Fn  A
)
6 ofcfval2.4 . . . 4  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
76fneq1d 5372 . . 3  |-  ( ph  ->  ( F  Fn  A  <->  ( x  e.  A  |->  B )  Fn  A ) )
85, 7mpbird 223 . 2  |-  ( ph  ->  F  Fn  A )
9 ofcfval2.1 . 2  |-  ( ph  ->  A  e.  V )
10 ofcfval2.2 . 2  |-  ( ph  ->  C  e.  W )
116, 1fvmpt2d 23222 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
128, 9, 10, 11ofcfval 23657 1  |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( x  e.  A  |->  ( B R C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   A.wral 2577    e. cmpt 4114    Fn wfn 5287  (class class class)co 5900  ∘𝑓/𝑐cofc 23654
This theorem is referenced by:  coinflippv  23915
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-ofc 23655
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