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Theorem ofrval 6573
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  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 2463 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
7 eqidd 2463 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
81, 2, 3, 4, 5, 6, 7ofrfval 6571 . . . . 5  |-  ( ph  ->  ( F  oR R G  <->  A. x  e.  S  ( F `  x ) R ( G `  x ) ) )
98biimpa 491 . . . 4  |-  ( (
ph  /\  F  oR R G )  ->  A. x  e.  S  ( F `  x ) R ( G `  x ) )
10 fveq2 5892 . . . . . 6  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
11 fveq2 5892 . . . . . 6  |-  ( x  =  X  ->  ( G `  x )  =  ( G `  X ) )
1210, 11breq12d 4431 . . . . 5  |-  ( x  =  X  ->  (
( F `  x
) R ( G `
 x )  <->  ( F `  X ) R ( G `  X ) ) )
1312rspccv 3159 . . . 4  |-  ( A. x  e.  S  ( F `  x ) R ( G `  x )  ->  ( X  e.  S  ->  ( F `  X ) R ( G `  X ) ) )
149, 13syl 17 . . 3  |-  ( (
ph  /\  F  oR R G )  ->  ( X  e.  S  ->  ( F `  X ) R ( G `  X ) ) )
15143impia 1212 . 2  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  ( F `  X
) R ( G `
 X ) )
16 simp1 1014 . . 3  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  ph )
17 inss1 3664 . . . . 5  |-  ( A  i^i  B )  C_  A
185, 17eqsstr3i 3475 . . . 4  |-  S  C_  A
19 simp3 1016 . . . 4  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  X  e.  S )
2018, 19sseldi 3442 . . 3  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  X  e.  A )
21 ofval.6 . . 3  |-  ( (
ph  /\  X  e.  A )  ->  ( F `  X )  =  C )
2216, 20, 21syl2anc 671 . 2  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  ( F `  X
)  =  C )
23 inss2 3665 . . . . 5  |-  ( A  i^i  B )  C_  B
245, 23eqsstr3i 3475 . . . 4  |-  S  C_  B
2524, 19sseldi 3442 . . 3  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  X  e.  B )
26 ofval.7 . . 3  |-  ( (
ph  /\  X  e.  B )  ->  ( G `  X )  =  D )
2716, 25, 26syl2anc 671 . 2  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  ( G `  X
)  =  D )
2815, 22, 273brtr3d 4448 1  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  C R D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898   A.wral 2749    i^i cin 3415   class class class wbr 4418    Fn wfn 5600   ` cfv 5605    oRcofr 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pr 4656
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-ofr 6564
This theorem is referenced by:  itg1le  22727  gsumle  28593  ftc1anclem5  32067
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