MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ofrval Unicode version

Theorem ofrval 6256
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 2390 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
7 eqidd 2390 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
81, 2, 3, 4, 5, 6, 7ofrfval 6254 . . . . 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 5670 . . . . . 6  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
11 fveq2 5670 . . . . . 6  |-  ( x  =  X  ->  ( G `  x )  =  ( G `  X ) )
1210, 11breq12d 4168 . . . . 5  |-  ( x  =  X  ->  (
( F `  x
) R ( G `
 x )  <->  ( F `  X ) R ( G `  X ) ) )
1312rspccv 2994 . . . 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 3506 . . . . 5  |-  ( A  i^i  B )  C_  A
185, 17eqsstr3i 3324 . . . 4  |-  S  C_  A
19 simp3 959 . . . 4  |-  ( (
ph  /\  F  o R R G  /\  X  e.  S )  ->  X  e.  S )
2018, 19sseldi 3291 . . 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 3507 . . . . 5  |-  ( A  i^i  B )  C_  B
245, 23eqsstr3i 3324 . . . 4  |-  S  C_  B
2524, 19sseldi 3291 . . 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 4184 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 1649    e. wcel 1717   A.wral 2651    i^i cin 3264   class class class wbr 4155    Fn wfn 5391   ` cfv 5396    o Rcofr 6245
This theorem is referenced by:  itg1le  19474
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ofr 6247
  Copyright terms: Public domain W3C validator