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Theorem eqfnfv3 5831
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 5830 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( F  =  G  <-> 
( A  =  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) ) )
2 eqss 3365 . . . . 5  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
3 ancom 439 . . . . 5  |-  ( ( A  C_  B  /\  B  C_  A )  <->  ( B  C_  A  /\  A  C_  B ) )
42, 3bitri 242 . . . 4  |-  ( A  =  B  <->  ( B  C_  A  /\  A  C_  B ) )
54anbi1i 678 . . 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 632 . . . 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 3340 . . . . . . 7  |-  ( A 
C_  B  <->  A. x  e.  A  x  e.  B )
87anbi1i 678 . . . . . 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 2840 . . . . . 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 245 . . . . 5  |-  ( ( A  C_  B  /\  A. x  e.  A  ( F `  x )  =  ( G `  x ) )  <->  A. x  e.  A  ( x  e.  B  /\  ( F `  x )  =  ( G `  x ) ) )
1110anbi2i 677 . . . 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 242 . . 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 242 . 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 254 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322    Fn wfn 5451   ` cfv 5456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-fv 5464
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