MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ofrval Structured version   Unicode version

Theorem ofrval 6549
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 2458 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
7 eqidd 2458 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  ( G `  x ) )
81, 2, 3, 4, 5, 6, 7ofrfval 6547 . . . . 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 5872 . . . . . 6  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
11 fveq2 5872 . . . . . 6  |-  ( x  =  X  ->  ( G `  x )  =  ( G `  X ) )
1210, 11breq12d 4469 . . . . 5  |-  ( x  =  X  ->  (
( F `  x
) R ( G `
 x )  <->  ( F `  X ) R ( G `  X ) ) )
1312rspccv 3207 . . . 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 1193 . 2  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  ( F `  X
) R ( G `
 X ) )
16 simp1 996 . . 3  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  ph )
17 inss1 3714 . . . . 5  |-  ( A  i^i  B )  C_  A
185, 17eqsstr3i 3530 . . . 4  |-  S  C_  A
19 simp3 998 . . . 4  |-  ( (
ph  /\  F  oR R G  /\  X  e.  S )  ->  X  e.  S )
2018, 19sseldi 3497 . . 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 3715 . . . . 5  |-  ( A  i^i  B )  C_  B
245, 23eqsstr3i 3530 . . . 4  |-  S  C_  B
2524, 19sseldi 3497 . . 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 4485 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 973    = wceq 1395    e. wcel 1819   A.wral 2807    i^i cin 3470   class class class wbr 4456    Fn wfn 5589   ` cfv 5594    oRcofr 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ofr 6540
This theorem is referenced by:  itg1le  22245  gsumle  27922  ftc1anclem5  30256
  Copyright terms: Public domain W3C validator