MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqfnfv3 Structured version   Visualization version   Unicode version

Theorem eqfnfv3 5978
Description: Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
eqfnfv3  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G  <-> 
( B  C_  A  /\  A. x  e.  A  ( x  e.  B  /\  ( F `  x
)  =  ( G `
 x ) ) ) ) )
Distinct variable groups:    x, A    x, F    x, G    x, B

Proof of Theorem eqfnfv3
StepHypRef Expression
1 eqfnfv2 5977 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G  <-> 
( A  =  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) ) )
2 eqss 3447 . . . . 5  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
3 ancom 452 . . . . 5  |-  ( ( A  C_  B  /\  B  C_  A )  <->  ( B  C_  A  /\  A  C_  B ) )
42, 3bitri 253 . . . 4  |-  ( A  =  B  <->  ( B  C_  A  /\  A  C_  B ) )
54anbi1i 701 . . 3  |-  ( ( A  =  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) )  <->  ( ( B  C_  A  /\  A  C_  B )  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
6 anass 655 . . . 4  |-  ( ( ( B  C_  A  /\  A  C_  B )  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) )  <->  ( B  C_  A  /\  ( A  C_  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) ) )
7 dfss3 3422 . . . . . . 7  |-  ( A 
C_  B  <->  A. x  e.  A  x  e.  B )
87anbi1i 701 . . . . . 6  |-  ( ( A  C_  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) )  <->  ( A. x  e.  A  x  e.  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
9 r19.26 2917 . . . . . 6  |-  ( A. x  e.  A  (
x  e.  B  /\  ( F `  x )  =  ( G `  x ) )  <->  ( A. x  e.  A  x  e.  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
108, 9bitr4i 256 . . . . 5  |-  ( ( A  C_  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) )  <->  A. x  e.  A  ( x  e.  B  /\  ( F `  x )  =  ( G `  x ) ) )
1110anbi2i 700 . . . 4  |-  ( ( B  C_  A  /\  ( A  C_  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )  <-> 
( B  C_  A  /\  A. x  e.  A  ( x  e.  B  /\  ( F `  x
)  =  ( G `
 x ) ) ) )
126, 11bitri 253 . . 3  |-  ( ( ( B  C_  A  /\  A  C_  B )  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) )  <->  ( B  C_  A  /\  A. x  e.  A  ( x  e.  B  /\  ( F `
 x )  =  ( G `  x
) ) ) )
135, 12bitri 253 . 2  |-  ( ( A  =  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) )  <->  ( B  C_  A  /\  A. x  e.  A  ( x  e.  B  /\  ( F `  x )  =  ( G `  x ) ) ) )
141, 13syl6bb 265 1  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G  <-> 
( B  C_  A  /\  A. x  e.  A  ( x  e.  B  /\  ( F `  x
)  =  ( G `
 x ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737    C_ wss 3404    Fn wfn 5577   ` cfv 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-fv 5590
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator