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Theorem ofcfeqd2 26479
Description: Equality theorem for function/constant operation value. (Contributed by Thierry Arnoux, 31-Jan-2017.)
Hypotheses
Ref Expression
ofcfeqd2.1  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  e.  B )
ofcfeqd2.2  |-  ( (
ph  /\  y  e.  B )  ->  (
y R C )  =  ( y P C ) )
ofcfeqd2.3  |-  ( ph  ->  F  Fn  A )
ofcfeqd2.4  |-  ( ph  ->  A  e.  V )
ofcfeqd2.5  |-  ( ph  ->  C  e.  W )
Assertion
Ref Expression
ofcfeqd2  |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( F𝑓/𝑐 P C ) )
Distinct variable groups:    x, y, C    x, F, y    x, P, y    x, R, y    ph, x, y    y, B
Allowed substitution hints:    A( x, y)    B( x)    V( x, y)    W( x, y)

Proof of Theorem ofcfeqd2
StepHypRef Expression
1 ofcfeqd2.1 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  e.  B )
2 ofcfeqd2.2 . . . . . 6  |-  ( (
ph  /\  y  e.  B )  ->  (
y R C )  =  ( y P C ) )
32ralrimiva 2797 . . . . 5  |-  ( ph  ->  A. y  e.  B  ( y R C )  =  ( y P C ) )
43adantr 462 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  A. y  e.  B  ( y R C )  =  ( y P C ) )
5 oveq1 6097 . . . . . 6  |-  ( y  =  ( F `  x )  ->  (
y R C )  =  ( ( F `
 x ) R C ) )
6 oveq1 6097 . . . . . 6  |-  ( y  =  ( F `  x )  ->  (
y P C )  =  ( ( F `
 x ) P C ) )
75, 6eqeq12d 2455 . . . . 5  |-  ( y  =  ( F `  x )  ->  (
( y R C )  =  ( y P C )  <->  ( ( F `  x ) R C )  =  ( ( F `  x
) P C ) ) )
87rspcva 3068 . . . 4  |-  ( ( ( F `  x
)  e.  B  /\  A. y  e.  B  ( y R C )  =  ( y P C ) )  -> 
( ( F `  x ) R C )  =  ( ( F `  x ) P C ) )
91, 4, 8syl2anc 656 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
( F `  x
) R C )  =  ( ( F `
 x ) P C ) )
109mpteq2dva 4375 . 2  |-  ( ph  ->  ( x  e.  A  |->  ( ( F `  x ) R C ) )  =  ( x  e.  A  |->  ( ( F `  x
) P C ) ) )
11 ofcfeqd2.3 . . 3  |-  ( ph  ->  F  Fn  A )
12 ofcfeqd2.4 . . 3  |-  ( ph  ->  A  e.  V )
13 ofcfeqd2.5 . . 3  |-  ( ph  ->  C  e.  W )
14 eqidd 2442 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
1511, 12, 13, 14ofcfval 26476 . 2  |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( x  e.  A  |->  ( ( F `  x
) R C ) ) )
1611, 12, 13, 14ofcfval 26476 . 2  |-  ( ph  ->  ( F𝑓/𝑐 P C )  =  ( x  e.  A  |->  ( ( F `  x
) P C ) ) )
1710, 15, 163eqtr4d 2483 1  |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( F𝑓/𝑐 P C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713    e. cmpt 4347    Fn wfn 5410   ` cfv 5415  (class class class)co 6090  ∘𝑓/𝑐cofc 26473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-ofc 26474
This theorem is referenced by:  coinfliplem  26791
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