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Theorem ofrval 6088
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 2284 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
7 eqidd 2284 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
81, 2, 3, 4, 5, 6, 7ofrfval 6086 . . . . 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 5525 . . . . . 6  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
11 fveq2 5525 . . . . . 6  |-  ( x  =  X  ->  ( G `  x )  =  ( G `  X ) )
1210, 11breq12d 4036 . . . . 5  |-  ( x  =  X  ->  (
( F `  x
) R ( G `
 x )  <->  ( F `  X ) R ( G `  X ) ) )
1312rspccv 2881 . . . 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 3389 . . . . 5  |-  ( A  i^i  B )  C_  A
185, 17eqsstr3i 3209 . . . 4  |-  S  C_  A
19 simp3 957 . . . 4  |-  ( (
ph  /\  F  o R R G  /\  X  e.  S )  ->  X  e.  S )
2018, 19sseldi 3178 . . 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 3390 . . . . 5  |-  ( A  i^i  B )  C_  B
245, 23eqsstr3i 3209 . . . 4  |-  S  C_  B
2524, 19sseldi 3178 . . 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 4052 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 1623    e. wcel 1684   A.wral 2543    i^i cin 3151   class class class wbr 4023    Fn wfn 5250   ` cfv 5255    o Rcofr 6077
This theorem is referenced by:  itg1le  19068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ofr 6079
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