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Theorem ofcfval2 28080
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.  X )
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)    X( x)

Proof of Theorem ofcfval2
StepHypRef Expression
1 ofcfval2.3 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  X )
21ralrimiva 2857 . . . 4  |-  ( ph  ->  A. x  e.  A  B  e.  X )
3 eqid 2443 . . . . 5  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
43fnmpt 5697 . . . 4  |-  ( A. x  e.  A  B  e.  X  ->  ( 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 5661 . . 3  |-  ( ph  ->  ( F  Fn  A  <->  ( x  e.  A  |->  B )  Fn  A ) )
85, 7mpbird 232 . 2  |-  ( ph  ->  F  Fn  A )
9 ofcfval2.1 . 2  |-  ( ph  ->  A  e.  V )
10 ofcfval2.2 . 2  |-  ( ph  ->  C  e.  W )
116, 1fvmpt2d 5950 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
128, 9, 10, 11ofcfval 28074 1  |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( x  e.  A  |->  ( B R C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793    |-> cmpt 4495    Fn wfn 5573  (class class class)co 6281  ∘𝑓/𝑐cofc 28071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-ofc 28072
This theorem is referenced by:  coinflippv  28399  ofcs1  28477
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