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Theorem ofrval 5946
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 2116 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
7 eqidd 2116 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
81, 2, 3, 4, 5, 6, 7ofrfval 5944 . . . . 5  |-  ( ph  ->  ( F  oR R G  <->  A. x  e.  S  ( F `  x ) R ( G `  x ) ) )
98biimpa 292 . . . 4  |-  ( (
ph  /\  F  oR R G )  ->  A. x  e.  S  ( F `  x ) R ( G `  x ) )
10 fveq2 5375 . . . . . 6  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
11 fveq2 5375 . . . . . 6  |-  ( x  =  X  ->  ( G `  x )  =  ( G `  X ) )
1210, 11breq12d 3908 . . . . 5  |-  ( x  =  X  ->  (
( F `  x
) R ( G `
 x )  <->  ( F `  X ) R ( G `  X ) ) )
1312rspccv 2757 . . . 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 1161 . 2  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  ( F `  X
) R ( G `
 X ) )
16 simp1 964 . . 3  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  ph )
17 inss1 3262 . . . . 5  |-  ( A  i^i  B )  C_  A
185, 17eqsstrri 3096 . . . 4  |-  S  C_  A
19 simp3 966 . . . 4  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  X  e.  S )
2018, 19sseldi 3061 . . 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 406 . 2  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  ( F `  X
)  =  C )
23 inss2 3263 . . . . 5  |-  ( A  i^i  B )  C_  B
245, 23eqsstrri 3096 . . . 4  |-  S  C_  B
2524, 19sseldi 3061 . . 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 406 . 2  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  ( G `  X
)  =  D )
2815, 22, 273brtr3d 3924 1  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  C R D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 945    = wceq 1314    e. wcel 1463   A.wral 2390    i^i cin 3036   class class class wbr 3895    Fn wfn 5076   ` cfv 5081    oRcofr 5935
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-ofr 5937
This theorem is referenced by: (None)
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