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Theorem ofrval 6307
Description: Exhibit a function relation at a point. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
offval.1  |-  ( ph  ->  F  Fn  A )
offval.2  |-  ( ph  ->  G  Fn  B )
offval.3  |-  ( ph  ->  A  e.  V )
offval.4  |-  ( ph  ->  B  e.  W )
offval.5  |-  ( A  i^i  B )  =  S
ofval.6  |-  ( (
ph  /\  X  e.  A )  ->  ( F `  X )  =  C )
ofval.7  |-  ( (
ph  /\  X  e.  B )  ->  ( G `  X )  =  D )
Assertion
Ref Expression
ofrval  |-  ( (
ph  /\  F  o R R G  /\  X  e.  S )  ->  C R D )

Proof of Theorem ofrval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 offval.1 . . . . . 6  |-  ( ph  ->  F  Fn  A )
2 offval.2 . . . . . 6  |-  ( ph  ->  G  Fn  B )
3 offval.3 . . . . . 6  |-  ( ph  ->  A  e.  V )
4 offval.4 . . . . . 6  |-  ( ph  ->  B  e.  W )
5 offval.5 . . . . . 6  |-  ( A  i^i  B )  =  S
6 eqidd 2436 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
7 eqidd 2436 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
81, 2, 3, 4, 5, 6, 7ofrfval 6305 . . . . 5  |-  ( ph  ->  ( F  o R R G  <->  A. x  e.  S  ( F `  x ) R ( G `  x ) ) )
98biimpa 471 . . . 4  |-  ( (
ph  /\  F  o R R G )  ->  A. x  e.  S  ( F `  x ) R ( G `  x ) )
10 fveq2 5720 . . . . . 6  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
11 fveq2 5720 . . . . . 6  |-  ( x  =  X  ->  ( G `  x )  =  ( G `  X ) )
1210, 11breq12d 4217 . . . . 5  |-  ( x  =  X  ->  (
( F `  x
) R ( G `
 x )  <->  ( F `  X ) R ( G `  X ) ) )
1312rspccv 3041 . . . 4  |-  ( A. x  e.  S  ( F `  x ) R ( G `  x )  ->  ( X  e.  S  ->  ( F `  X ) R ( G `  X ) ) )
149, 13syl 16 . . 3  |-  ( (
ph  /\  F  o R R G )  -> 
( X  e.  S  ->  ( F `  X
) R ( G `
 X ) ) )
15143impia 1150 . 2  |-  ( (
ph  /\  F  o R R G  /\  X  e.  S )  ->  ( F `  X ) R ( G `  X ) )
16 simp1 957 . . 3  |-  ( (
ph  /\  F  o R R G  /\  X  e.  S )  ->  ph )
17 inss1 3553 . . . . 5  |-  ( A  i^i  B )  C_  A
185, 17eqsstr3i 3371 . . . 4  |-  S  C_  A
19 simp3 959 . . . 4  |-  ( (
ph  /\  F  o R R G  /\  X  e.  S )  ->  X  e.  S )
2018, 19sseldi 3338 . . 3  |-  ( (
ph  /\  F  o R R G  /\  X  e.  S )  ->  X  e.  A )
21 ofval.6 . . 3  |-  ( (
ph  /\  X  e.  A )  ->  ( F `  X )  =  C )
2216, 20, 21syl2anc 643 . 2  |-  ( (
ph  /\  F  o R R G  /\  X  e.  S )  ->  ( F `  X )  =  C )
23 inss2 3554 . . . . 5  |-  ( A  i^i  B )  C_  B
245, 23eqsstr3i 3371 . . . 4  |-  S  C_  B
2524, 19sseldi 3338 . . 3  |-  ( (
ph  /\  F  o R R G  /\  X  e.  S )  ->  X  e.  B )
26 ofval.7 . . 3  |-  ( (
ph  /\  X  e.  B )  ->  ( G `  X )  =  D )
2716, 25, 26syl2anc 643 . 2  |-  ( (
ph  /\  F  o R R G  /\  X  e.  S )  ->  ( G `  X )  =  D )
2815, 22, 273brtr3d 4233 1  |-  ( (
ph  /\  F  o R R G  /\  X  e.  S )  ->  C R D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697    i^i cin 3311   class class class wbr 4204    Fn wfn 5441   ` cfv 5446    o Rcofr 6296
This theorem is referenced by:  itg1le  19597  ftc1anclem5  26274
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-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-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-ofr 6298
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