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Theorem ofrval 5774
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
ofrval.6  |-  ( (
ph  /\  X  e.  A )  ->  ( F `  X )  =  C )
ofrval.7  |-  ( (
ph  /\  X  e.  B )  ->  ( G `  X )  =  D )
Assertion
Ref Expression
ofrval  |-  ( (
ph  /\  F  oR 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 2084 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
7 eqidd 2084 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
81, 2, 3, 4, 5, 6, 7ofrfval 5772 . . . . 5  |-  ( ph  ->  ( F  oR R G  <->  A. x  e.  S  ( F `  x ) R ( G `  x ) ) )
98biimpa 290 . . . 4  |-  ( (
ph  /\  F  oR R G )  ->  A. x  e.  S  ( F `  x ) R ( G `  x ) )
10 fveq2 5230 . . . . . 6  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
11 fveq2 5230 . . . . . 6  |-  ( x  =  X  ->  ( G `  x )  =  ( G `  X ) )
1210, 11breq12d 3818 . . . . 5  |-  ( x  =  X  ->  (
( F `  x
) R ( G `
 x )  <->  ( F `  X ) R ( G `  X ) ) )
1312rspccv 2707 . . . 4  |-  ( A. x  e.  S  ( F `  x ) R ( G `  x )  ->  ( X  e.  S  ->  ( F `  X ) R ( G `  X ) ) )
149, 13syl 14 . . 3  |-  ( (
ph  /\  F  oR R G )  ->  ( X  e.  S  ->  ( F `  X ) R ( G `  X ) ) )
15143impia 1136 . 2  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  ( F `  X
) R ( G `
 X ) )
16 simp1 939 . . 3  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  ph )
17 inss1 3202 . . . . 5  |-  ( A  i^i  B )  C_  A
185, 17eqsstr3i 3039 . . . 4  |-  S  C_  A
19 simp3 941 . . . 4  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  X  e.  S )
2018, 19sseldi 3006 . . 3  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  X  e.  A )
21 ofrval.6 . . 3  |-  ( (
ph  /\  X  e.  A )  ->  ( F `  X )  =  C )
2216, 20, 21syl2anc 403 . 2  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  ( F `  X
)  =  C )
23 inss2 3203 . . . . 5  |-  ( A  i^i  B )  C_  B
245, 23eqsstr3i 3039 . . . 4  |-  S  C_  B
2524, 19sseldi 3006 . . 3  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  X  e.  B )
26 ofrval.7 . . 3  |-  ( (
ph  /\  X  e.  B )  ->  ( G `  X )  =  D )
2716, 25, 26syl2anc 403 . 2  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  ( G `  X
)  =  D )
2815, 22, 273brtr3d 3834 1  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  C R D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 920    = wceq 1285    e. wcel 1434   A.wral 2353    i^i cin 2981   class class class wbr 3805    Fn wfn 4947   ` cfv 4952    oRcofr 5763
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ofr 5765
This theorem is referenced by: (None)
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