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Theorem ofcfeqd2 28284
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 2810 . . . . 5  |-  ( ph  ->  A. y  e.  B  ( y R C )  =  ( y P C ) )
43adantr 463 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  A. y  e.  B  ( y R C )  =  ( y P C ) )
5 oveq1 6225 . . . . . 6  |-  ( y  =  ( F `  x )  ->  (
y R C )  =  ( ( F `
 x ) R C ) )
6 oveq1 6225 . . . . . 6  |-  ( y  =  ( F `  x )  ->  (
y P C )  =  ( ( F `
 x ) P C ) )
75, 6eqeq12d 2418 . . . . 5  |-  ( y  =  ( F `  x )  ->  (
( y R C )  =  ( y P C )  <->  ( ( F `  x ) R C )  =  ( ( F `  x
) P C ) ) )
87rspcva 3150 . . . 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 659 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
( F `  x
) R C )  =  ( ( F `
 x ) P C ) )
109mpteq2dva 4470 . 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 2397 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
1511, 12, 13, 14ofcfval 28281 . 2  |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( x  e.  A  |->  ( ( F `  x
) R C ) ) )
1611, 12, 13, 14ofcfval 28281 . 2  |-  ( ph  ->  ( F𝑓/𝑐 P C )  =  ( x  e.  A  |->  ( ( F `  x
) P C ) ) )
1710, 15, 163eqtr4d 2447 1  |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( F𝑓/𝑐 P C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1836   A.wral 2746    |-> cmpt 4442    Fn wfn 5508   ` cfv 5513  (class class class)co 6218  ∘𝑓/𝑐cofc 28278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pr 4618
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-ofc 28279
This theorem is referenced by:  coinfliplem  28640
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