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Theorem ofrval 6351
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 2444 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
7 eqidd 2444 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
81, 2, 3, 4, 5, 6, 7ofrfval 6349 . . . . 5  |-  ( ph  ->  ( F  oR R G  <->  A. x  e.  S  ( F `  x ) R ( G `  x ) ) )
98biimpa 484 . . . 4  |-  ( (
ph  /\  F  oR R G )  ->  A. x  e.  S  ( F `  x ) R ( G `  x ) )
10 fveq2 5712 . . . . . 6  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
11 fveq2 5712 . . . . . 6  |-  ( x  =  X  ->  ( G `  x )  =  ( G `  X ) )
1210, 11breq12d 4326 . . . . 5  |-  ( x  =  X  ->  (
( F `  x
) R ( G `
 x )  <->  ( F `  X ) R ( G `  X ) ) )
1312rspccv 3091 . . . 4  |-  ( A. x  e.  S  ( F `  x ) R ( G `  x )  ->  ( X  e.  S  ->  ( F `  X ) R ( G `  X ) ) )
149, 13syl 16 . . 3  |-  ( (
ph  /\  F  oR R G )  ->  ( X  e.  S  ->  ( F `  X ) R ( G `  X ) ) )
15143impia 1184 . 2  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  ( F `  X
) R ( G `
 X ) )
16 simp1 988 . . 3  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  ph )
17 inss1 3591 . . . . 5  |-  ( A  i^i  B )  C_  A
185, 17eqsstr3i 3408 . . . 4  |-  S  C_  A
19 simp3 990 . . . 4  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  X  e.  S )
2018, 19sseldi 3375 . . 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 661 . 2  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  ( F `  X
)  =  C )
23 inss2 3592 . . . . 5  |-  ( A  i^i  B )  C_  B
245, 23eqsstr3i 3408 . . . 4  |-  S  C_  B
2524, 19sseldi 3375 . . 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 661 . 2  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  ( G `  X
)  =  D )
2815, 22, 273brtr3d 4342 1  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  C R D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2736    i^i cin 3348   class class class wbr 4313    Fn wfn 5434   ` cfv 5439    oRcofr 6340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ofr 6342
This theorem is referenced by:  itg1le  21213  gsumle  26268  ftc1anclem5  28497
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