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Theorem eqfnung2 25118
Description: If a family of sets  A indexed by  I covers the common domain  B of two functions  F and  G, the restrictions of  F and  G to  ( A  i^i  B ) are equal iff  F  =  G. Compare eqfnun 26387. (Contributed by FL, 5-Nov-2011.)
Assertion
Ref Expression
eqfnung2  |-  ( ( B  C_  U_ i  e.  I  A  /\  F  Fn  B  /\  G  Fn  B )  ->  ( A. i  e.  I 
( F  |`  A )  =  ( G  |`  A )  <->  F  =  G ) )
Distinct variable groups:    i, F    i, G
Allowed substitution hints:    A( i)    B( i)    I( i)

Proof of Theorem eqfnung2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqidd 2284 . . . 4  |-  ( ( ( B  C_  U_ i  e.  I  A  /\  F  Fn  B  /\  G  Fn  B )  /\  A. i  e.  I 
( F  |`  A )  =  ( G  |`  A ) )  ->  B  =  B )
2 ssel 3174 . . . . . . . . . . 11  |-  ( B 
C_  U_ i  e.  I  A  ->  ( x  e.  B  ->  x  e.  U_ i  e.  I  A ) )
3 eliun 3909 . . . . . . . . . . . 12  |-  ( x  e.  U_ i  e.  I  A  <->  E. i  e.  I  x  e.  A )
4 r19.29 2683 . . . . . . . . . . . . . . . . 17  |-  ( ( A. i  e.  I 
( F  |`  A )  =  ( G  |`  A )  /\  E. i  e.  I  x  e.  A )  ->  E. i  e.  I  ( ( F  |`  A )  =  ( G  |`  A )  /\  x  e.  A
) )
5 fveq1 5524 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( F  |`  A )  =  ( G  |`  A )  ->  (
( F  |`  A ) `
 x )  =  ( ( G  |`  A ) `  x
) )
6 fvres 5542 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  e.  A  ->  (
( F  |`  A ) `
 x )  =  ( F `  x
) )
7 fvres 5542 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  e.  A  ->  (
( G  |`  A ) `
 x )  =  ( G `  x
) )
8 eqeq12 2295 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( F  |`  A ) `  x
)  =  ( F `
 x )  /\  ( ( G  |`  A ) `  x
)  =  ( G `
 x ) )  ->  ( ( ( F  |`  A ) `  x )  =  ( ( G  |`  A ) `
 x )  <->  ( F `  x )  =  ( G `  x ) ) )
98biimpd 198 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( F  |`  A ) `  x
)  =  ( F `
 x )  /\  ( ( G  |`  A ) `  x
)  =  ( G `
 x ) )  ->  ( ( ( F  |`  A ) `  x )  =  ( ( G  |`  A ) `
 x )  -> 
( F `  x
)  =  ( G `
 x ) ) )
106, 7, 9syl2anc 642 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( x  e.  A  ->  (
( ( F  |`  A ) `  x
)  =  ( ( G  |`  A ) `  x )  ->  ( F `  x )  =  ( G `  x ) ) )
115, 10mpan9 455 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( F  |`  A )  =  ( G  |`  A )  /\  x  e.  A )  ->  ( F `  x )  =  ( G `  x ) )
1211rexlimivw 2663 . . . . . . . . . . . . . . . . . . . . 21  |-  ( E. i  e.  I  ( ( F  |`  A )  =  ( G  |`  A )  /\  x  e.  A )  ->  ( F `  x )  =  ( G `  x ) )
1312a1d 22 . . . . . . . . . . . . . . . . . . . 20  |-  ( E. i  e.  I  ( ( F  |`  A )  =  ( G  |`  A )  /\  x  e.  A )  ->  ( B  C_  U_ i  e.  I  A  ->  ( F `  x )  =  ( G `  x ) ) )
1413a1i 10 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  B  ->  ( E. i  e.  I 
( ( F  |`  A )  =  ( G  |`  A )  /\  x  e.  A
)  ->  ( B  C_ 
U_ i  e.  I  A  ->  ( F `  x )  =  ( G `  x ) ) ) )
1514a1ii 24 . . . . . . . . . . . . . . . . . 18  |-  ( F  Fn  B  ->  ( G  Fn  B  ->  ( x  e.  B  -> 
( E. i  e.  I  ( ( F  |`  A )  =  ( G  |`  A )  /\  x  e.  A
)  ->  ( B  C_ 
U_ i  e.  I  A  ->  ( F `  x )  =  ( G `  x ) ) ) ) ) )
1615com4r 80 . . . . . . . . . . . . . . . . 17  |-  ( E. i  e.  I  ( ( F  |`  A )  =  ( G  |`  A )  /\  x  e.  A )  ->  ( F  Fn  B  ->  ( G  Fn  B  -> 
( x  e.  B  ->  ( B  C_  U_ i  e.  I  A  ->  ( F `  x )  =  ( G `  x ) ) ) ) ) )
174, 16syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( A. i  e.  I 
( F  |`  A )  =  ( G  |`  A )  /\  E. i  e.  I  x  e.  A )  ->  ( F  Fn  B  ->  ( G  Fn  B  -> 
( x  e.  B  ->  ( B  C_  U_ i  e.  I  A  ->  ( F `  x )  =  ( G `  x ) ) ) ) ) )
1817ex 423 . . . . . . . . . . . . . . 15  |-  ( A. i  e.  I  ( F  |`  A )  =  ( G  |`  A )  ->  ( E. i  e.  I  x  e.  A  ->  ( F  Fn  B  ->  ( G  Fn  B  ->  ( x  e.  B  ->  ( B  C_ 
U_ i  e.  I  A  ->  ( F `  x )  =  ( G `  x ) ) ) ) ) ) )
1918com4t 79 . . . . . . . . . . . . . 14  |-  ( F  Fn  B  ->  ( G  Fn  B  ->  ( A. i  e.  I 
( F  |`  A )  =  ( G  |`  A )  ->  ( E. i  e.  I  x  e.  A  ->  ( x  e.  B  -> 
( B  C_  U_ i  e.  I  A  ->  ( F `  x )  =  ( G `  x ) ) ) ) ) ) )
20193imp 1145 . . . . . . . . . . . . 13  |-  ( ( F  Fn  B  /\  G  Fn  B  /\  A. i  e.  I  ( F  |`  A )  =  ( G  |`  A ) )  -> 
( E. i  e.  I  x  e.  A  ->  ( x  e.  B  ->  ( B  C_  U_ i  e.  I  A  ->  ( F `  x )  =  ( G `  x ) ) ) ) )
2120com3l 75 . . . . . . . . . . . 12  |-  ( E. i  e.  I  x  e.  A  ->  (
x  e.  B  -> 
( ( F  Fn  B  /\  G  Fn  B  /\  A. i  e.  I 
( F  |`  A )  =  ( G  |`  A ) )  -> 
( B  C_  U_ i  e.  I  A  ->  ( F `  x )  =  ( G `  x ) ) ) ) )
223, 21sylbi 187 . . . . . . . . . . 11  |-  ( x  e.  U_ i  e.  I  A  ->  (
x  e.  B  -> 
( ( F  Fn  B  /\  G  Fn  B  /\  A. i  e.  I 
( F  |`  A )  =  ( G  |`  A ) )  -> 
( B  C_  U_ i  e.  I  A  ->  ( F `  x )  =  ( G `  x ) ) ) ) )
232, 22syl6com 31 . . . . . . . . . 10  |-  ( x  e.  B  ->  ( B  C_  U_ i  e.  I  A  ->  (
x  e.  B  -> 
( ( F  Fn  B  /\  G  Fn  B  /\  A. i  e.  I 
( F  |`  A )  =  ( G  |`  A ) )  -> 
( B  C_  U_ i  e.  I  A  ->  ( F `  x )  =  ( G `  x ) ) ) ) ) )
2423pm2.43a 45 . . . . . . . . 9  |-  ( x  e.  B  ->  ( B  C_  U_ i  e.  I  A  ->  (
( F  Fn  B  /\  G  Fn  B  /\  A. i  e.  I 
( F  |`  A )  =  ( G  |`  A ) )  -> 
( B  C_  U_ i  e.  I  A  ->  ( F `  x )  =  ( G `  x ) ) ) ) )
2524com14 82 . . . . . . . 8  |-  ( B 
C_  U_ i  e.  I  A  ->  ( B  C_  U_ i  e.  I  A  ->  ( ( F  Fn  B  /\  G  Fn  B  /\  A. i  e.  I  ( F  |`  A )  =  ( G  |`  A )
)  ->  ( x  e.  B  ->  ( F `
 x )  =  ( G `  x
) ) ) ) )
2625pm2.43i 43 . . . . . . 7  |-  ( B 
C_  U_ i  e.  I  A  ->  ( ( F  Fn  B  /\  G  Fn  B  /\  A. i  e.  I  ( F  |`  A )  =  ( G  |`  A )
)  ->  ( x  e.  B  ->  ( F `
 x )  =  ( G `  x
) ) ) )
27263expd 1168 . . . . . 6  |-  ( B 
C_  U_ i  e.  I  A  ->  ( F  Fn  B  ->  ( G  Fn  B  ->  ( A. i  e.  I  ( F  |`  A )  =  ( G  |`  A )  ->  ( x  e.  B  ->  ( F `  x
)  =  ( G `
 x ) ) ) ) ) )
28273imp1 1164 . . . . 5  |-  ( ( ( B  C_  U_ i  e.  I  A  /\  F  Fn  B  /\  G  Fn  B )  /\  A. i  e.  I 
( F  |`  A )  =  ( G  |`  A ) )  -> 
( x  e.  B  ->  ( F `  x
)  =  ( G `
 x ) ) )
2928ralrimiv 2625 . . . 4  |-  ( ( ( B  C_  U_ i  e.  I  A  /\  F  Fn  B  /\  G  Fn  B )  /\  A. i  e.  I 
( F  |`  A )  =  ( G  |`  A ) )  ->  A. x  e.  B  ( F `  x )  =  ( G `  x ) )
30 3simpc 954 . . . . . 6  |-  ( ( B  C_  U_ i  e.  I  A  /\  F  Fn  B  /\  G  Fn  B )  ->  ( F  Fn  B  /\  G  Fn  B )
)
3130adantr 451 . . . . 5  |-  ( ( ( B  C_  U_ i  e.  I  A  /\  F  Fn  B  /\  G  Fn  B )  /\  A. i  e.  I 
( F  |`  A )  =  ( G  |`  A ) )  -> 
( F  Fn  B  /\  G  Fn  B
) )
32 eqfnfv2 5623 . . . . 5  |-  ( ( F  Fn  B  /\  G  Fn  B )  ->  ( F  =  G  <-> 
( B  =  B  /\  A. x  e.  B  ( F `  x )  =  ( G `  x ) ) ) )
3331, 32syl 15 . . . 4  |-  ( ( ( B  C_  U_ i  e.  I  A  /\  F  Fn  B  /\  G  Fn  B )  /\  A. i  e.  I 
( F  |`  A )  =  ( G  |`  A ) )  -> 
( F  =  G  <-> 
( B  =  B  /\  A. x  e.  B  ( F `  x )  =  ( G `  x ) ) ) )
341, 29, 33mpbir2and 888 . . 3  |-  ( ( ( B  C_  U_ i  e.  I  A  /\  F  Fn  B  /\  G  Fn  B )  /\  A. i  e.  I 
( F  |`  A )  =  ( G  |`  A ) )  ->  F  =  G )
3534ex 423 . 2  |-  ( ( B  C_  U_ i  e.  I  A  /\  F  Fn  B  /\  G  Fn  B )  ->  ( A. i  e.  I 
( F  |`  A )  =  ( G  |`  A )  ->  F  =  G ) )
36 reseq1 4949 . . 3  |-  ( F  =  G  ->  ( F  |`  A )  =  ( G  |`  A ) )
3736ralrimivw 2627 . 2  |-  ( F  =  G  ->  A. i  e.  I  ( F  |`  A )  =  ( G  |`  A )
)
3835, 37impbid1 194 1  |-  ( ( B  C_  U_ i  e.  I  A  /\  F  Fn  B  /\  G  Fn  B )  ->  ( A. i  e.  I 
( F  |`  A )  =  ( G  |`  A )  <->  F  =  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   U_ciun 3905    |` cres 4691    Fn wfn 5250   ` cfv 5255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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