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Theorem eqfnfv3 5649
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 5648 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G  <-> 
( A  =  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) ) )
2 eqss 3194 . . . . 5  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
3 ancom 266 . . . . 5  |-  ( ( A  C_  B  /\  B  C_  A )  <->  ( B  C_  A  /\  A  C_  B ) )
42, 3bitri 184 . . . 4  |-  ( A  =  B  <->  ( B  C_  A  /\  A  C_  B ) )
54anbi1i 458 . . 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 401 . . . 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 3169 . . . . . . 7  |-  ( A 
C_  B  <->  A. x  e.  A  x  e.  B )
87anbi1i 458 . . . . . 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 2620 . . . . . 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 187 . . . . 5  |-  ( ( A  C_  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) )  <->  A. x  e.  A  ( x  e.  B  /\  ( F `  x )  =  ( G `  x ) ) )
1110anbi2i 457 . . . 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 184 . . 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 184 . 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, 13bitrdi 196 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 104    <-> wb 105    = wceq 1364    e. wcel 2164   A.wral 2472    C_ wss 3153    Fn wfn 5241   ` cfv 5246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-iota 5207  df-fun 5248  df-fn 5249  df-fv 5254
This theorem is referenced by: (None)
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