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Theorem ofrval 6104
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 2297 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
7 eqidd 2297 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
81, 2, 3, 4, 5, 6, 7ofrfval 6102 . . . . 5  |-  ( ph  ->  ( F  o R R G  <->  A. x  e.  S  ( F `  x ) R ( G `  x ) ) )
98biimpa 470 . . . 4  |-  ( (
ph  /\  F  o R R G )  ->  A. x  e.  S  ( F `  x ) R ( G `  x ) )
10 fveq2 5541 . . . . . 6  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
11 fveq2 5541 . . . . . 6  |-  ( x  =  X  ->  ( G `  x )  =  ( G `  X ) )
1210, 11breq12d 4052 . . . . 5  |-  ( x  =  X  ->  (
( F `  x
) R ( G `
 x )  <->  ( F `  X ) R ( G `  X ) ) )
1312rspccv 2894 . . . 4  |-  ( A. x  e.  S  ( F `  x ) R ( G `  x )  ->  ( X  e.  S  ->  ( F `  X ) R ( G `  X ) ) )
149, 13syl 15 . . 3  |-  ( (
ph  /\  F  o R R G )  -> 
( X  e.  S  ->  ( F `  X
) R ( G `
 X ) ) )
15143impia 1148 . 2  |-  ( (
ph  /\  F  o R R G  /\  X  e.  S )  ->  ( F `  X ) R ( G `  X ) )
16 simp1 955 . . 3  |-  ( (
ph  /\  F  o R R G  /\  X  e.  S )  ->  ph )
17 inss1 3402 . . . . 5  |-  ( A  i^i  B )  C_  A
185, 17eqsstr3i 3222 . . . 4  |-  S  C_  A
19 simp3 957 . . . 4  |-  ( (
ph  /\  F  o R R G  /\  X  e.  S )  ->  X  e.  S )
2018, 19sseldi 3191 . . 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 642 . 2  |-  ( (
ph  /\  F  o R R G  /\  X  e.  S )  ->  ( F `  X )  =  C )
23 inss2 3403 . . . . 5  |-  ( A  i^i  B )  C_  B
245, 23eqsstr3i 3222 . . . 4  |-  S  C_  B
2524, 19sseldi 3191 . . 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 642 . 2  |-  ( (
ph  /\  F  o R R G  /\  X  e.  S )  ->  ( G `  X )  =  D )
2815, 22, 273brtr3d 4068 1  |-  ( (
ph  /\  F  o R R G  /\  X  e.  S )  ->  C R D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    i^i cin 3164   class class class wbr 4039    Fn wfn 5266   ` cfv 5271    o Rcofr 6093
This theorem is referenced by:  itg1le  19084
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ofr 6095
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