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Theorem eqfnfv3 5640
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 5639 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G  <-> 
( A  =  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) ) )
2 eqss 3207 . . . . 5  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
3 ancom 437 . . . . 5  |-  ( ( A  C_  B  /\  B  C_  A )  <->  ( B  C_  A  /\  A  C_  B ) )
42, 3bitri 240 . . . 4  |-  ( A  =  B  <->  ( B  C_  A  /\  A  C_  B ) )
54anbi1i 676 . . 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 630 . . . 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 3183 . . . . . . 7  |-  ( A 
C_  B  <->  A. x  e.  A  x  e.  B )
87anbi1i 676 . . . . . 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 2688 . . . . . 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 243 . . . . 5  |-  ( ( A  C_  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) )  <->  A. x  e.  A  ( x  e.  B  /\  ( F `  x )  =  ( G `  x ) ) )
1110anbi2i 675 . . . 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 240 . . 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 240 . 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 252 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165    Fn wfn 5266   ` cfv 5271
This theorem is referenced by:  prl2  25272  eqfnfv3OLD  26501
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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