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Theorem eqfnfv3 5320
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 5319 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G  <-> 
( A  =  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) ) )
2 eqss 3024 . . . . 5  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
3 ancom 262 . . . . 5  |-  ( ( A  C_  B  /\  B  C_  A )  <->  ( B  C_  A  /\  A  C_  B ) )
42, 3bitri 182 . . . 4  |-  ( A  =  B  <->  ( B  C_  A  /\  A  C_  B ) )
54anbi1i 446 . . 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 393 . . . 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 2999 . . . . . . 7  |-  ( A 
C_  B  <->  A. x  e.  A  x  e.  B )
87anbi1i 446 . . . . . 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 2490 . . . . . 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 185 . . . . 5  |-  ( ( A  C_  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) )  <->  A. x  e.  A  ( x  e.  B  /\  ( F `  x )  =  ( G `  x ) ) )
1110anbi2i 445 . . . 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 182 . . 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 182 . 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 194 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    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2353    C_ wss 2983    Fn wfn 4948   ` cfv 4953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2826  df-csb 2919  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-opab 3861  df-mpt 3862  df-id 4077  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-iota 4918  df-fun 4955  df-fn 4956  df-fv 4961
This theorem is referenced by: (None)
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