Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ofcfeqd2 Structured version   Unicode version

Theorem ofcfeqd2 26558
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 2814 . . . . 5  |-  ( ph  ->  A. y  e.  B  ( y R C )  =  ( y P C ) )
43adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  A. y  e.  B  ( y R C )  =  ( y P C ) )
5 oveq1 6113 . . . . . 6  |-  ( y  =  ( F `  x )  ->  (
y R C )  =  ( ( F `
 x ) R C ) )
6 oveq1 6113 . . . . . 6  |-  ( y  =  ( F `  x )  ->  (
y P C )  =  ( ( F `
 x ) P C ) )
75, 6eqeq12d 2457 . . . . 5  |-  ( y  =  ( F `  x )  ->  (
( y R C )  =  ( y P C )  <->  ( ( F `  x ) R C )  =  ( ( F `  x
) P C ) ) )
87rspcva 3086 . . . 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 661 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
( F `  x
) R C )  =  ( ( F `
 x ) P C ) )
109mpteq2dva 4393 . 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 2444 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
1511, 12, 13, 14ofcfval 26555 . 2  |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( x  e.  A  |->  ( ( F `  x
) R C ) ) )
1611, 12, 13, 14ofcfval 26555 . 2  |-  ( ph  ->  ( F𝑓/𝑐 P C )  =  ( x  e.  A  |->  ( ( F `  x
) P C ) ) )
1710, 15, 163eqtr4d 2485 1  |-  ( ph  ->  ( F𝑓/𝑐 R C )  =  ( F𝑓/𝑐 P C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2730    e. cmpt 4365    Fn wfn 5428   ` cfv 5433  (class class class)co 6106  ∘𝑓/𝑐cofc 26552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pr 4546
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-ofc 26553
This theorem is referenced by:  coinfliplem  26876
  Copyright terms: Public domain W3C validator