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Theorem ofcfval2 24479
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 2781 . . . 4  |-  ( ph  ->  A. x  e.  A  B  e.  W )
3 eqid 2435 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
43fnmpt 5563 . . . 4  |-  ( A. x  e.  A  B  e.  W  ->  ( x  e.  A  |->  B )  Fn  A )
52, 4syl 16 . . 3  |-  ( ph  ->  ( x  e.  A  |->  B )  Fn  A
)
6 ofcfval2.4 . . . 4  |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )
76fneq1d 5528 . . 3  |-  ( ph  ->  ( F  Fn  A  <->  ( x  e.  A  |->  B )  Fn  A ) )
85, 7mpbird 224 . 2  |-  ( ph  ->  F  Fn  A )
9 ofcfval2.1 . 2  |-  ( ph  ->  A  e.  V )
10 ofcfval2.2 . 2  |-  ( ph  ->  C  e.  W )
116, 1fvmpt2d 5806 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
128, 9, 10, 11ofcfval 24473 1  |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( x  e.  A  |->  ( B R C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    e. cmpt 4258    Fn wfn 5441  (class class class)co 6073  ∘𝑓/𝑐cofc 24470
This theorem is referenced by:  coinflippv  24733
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-ofc 24471
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