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Theorem 2f1fvneq 32097
Description: If two one-to-one functions are applied on different arguments, also the values are different. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
Assertion
Ref Expression
2f1fvneq  |-  ( ( ( E : D -1-1-> R  /\  F : C -1-1-> D )  /\  ( A  e.  C  /\  B  e.  C )  /\  A  =/=  B )  ->  (
( ( E `  ( F `  A ) )  =  X  /\  ( E `  ( F `
 B ) )  =  Y )  ->  X  =/=  Y ) )

Proof of Theorem 2f1fvneq
StepHypRef Expression
1 f1veqaeq 6167 . . . . 5  |-  ( ( F : C -1-1-> D  /\  ( A  e.  C  /\  B  e.  C
) )  ->  (
( F `  A
)  =  ( F `
 B )  ->  A  =  B )
)
21adantll 713 . . . 4  |-  ( ( ( E : D -1-1-> R  /\  F : C -1-1-> D )  /\  ( A  e.  C  /\  B  e.  C ) )  -> 
( ( F `  A )  =  ( F `  B )  ->  A  =  B ) )
32necon3ad 2677 . . 3  |-  ( ( ( E : D -1-1-> R  /\  F : C -1-1-> D )  /\  ( A  e.  C  /\  B  e.  C ) )  -> 
( A  =/=  B  ->  -.  ( F `  A )  =  ( F `  B ) ) )
433impia 1193 . 2  |-  ( ( ( E : D -1-1-> R  /\  F : C -1-1-> D )  /\  ( A  e.  C  /\  B  e.  C )  /\  A  =/=  B )  ->  -.  ( F `  A )  =  ( F `  B ) )
5 simpll 753 . . . . . . 7  |-  ( ( ( E : D -1-1-> R  /\  F : C -1-1-> D )  /\  ( A  e.  C  /\  B  e.  C ) )  ->  E : D -1-1-> R )
6 f1f 5787 . . . . . . . . . 10  |-  ( F : C -1-1-> D  ->  F : C --> D )
7 ffvelrn 6030 . . . . . . . . . . . 12  |-  ( ( F : C --> D  /\  A  e.  C )  ->  ( F `  A
)  e.  D )
8 ffvelrn 6030 . . . . . . . . . . . 12  |-  ( ( F : C --> D  /\  B  e.  C )  ->  ( F `  B
)  e.  D )
97, 8anim12dan 835 . . . . . . . . . . 11  |-  ( ( F : C --> D  /\  ( A  e.  C  /\  B  e.  C
) )  ->  (
( F `  A
)  e.  D  /\  ( F `  B )  e.  D ) )
109ex 434 . . . . . . . . . 10  |-  ( F : C --> D  -> 
( ( A  e.  C  /\  B  e.  C )  ->  (
( F `  A
)  e.  D  /\  ( F `  B )  e.  D ) ) )
116, 10syl 16 . . . . . . . . 9  |-  ( F : C -1-1-> D  -> 
( ( A  e.  C  /\  B  e.  C )  ->  (
( F `  A
)  e.  D  /\  ( F `  B )  e.  D ) ) )
1211adantl 466 . . . . . . . 8  |-  ( ( E : D -1-1-> R  /\  F : C -1-1-> D
)  ->  ( ( A  e.  C  /\  B  e.  C )  ->  ( ( F `  A )  e.  D  /\  ( F `  B
)  e.  D ) ) )
1312imp 429 . . . . . . 7  |-  ( ( ( E : D -1-1-> R  /\  F : C -1-1-> D )  /\  ( A  e.  C  /\  B  e.  C ) )  -> 
( ( F `  A )  e.  D  /\  ( F `  B
)  e.  D ) )
14 f1veqaeq 6167 . . . . . . 7  |-  ( ( E : D -1-1-> R  /\  ( ( F `  A )  e.  D  /\  ( F `  B
)  e.  D ) )  ->  ( ( E `  ( F `  A ) )  =  ( E `  ( F `  B )
)  ->  ( F `  A )  =  ( F `  B ) ) )
155, 13, 14syl2anc 661 . . . . . 6  |-  ( ( ( E : D -1-1-> R  /\  F : C -1-1-> D )  /\  ( A  e.  C  /\  B  e.  C ) )  -> 
( ( E `  ( F `  A ) )  =  ( E `
 ( F `  B ) )  -> 
( F `  A
)  =  ( F `
 B ) ) )
1615con3dimp 441 . . . . 5  |-  ( ( ( ( E : D -1-1-> R  /\  F : C -1-1-> D )  /\  ( A  e.  C  /\  B  e.  C
) )  /\  -.  ( F `  A )  =  ( F `  B ) )  ->  -.  ( E `  ( F `  A )
)  =  ( E `
 ( F `  B ) ) )
17 eqeq12 2486 . . . . . . 7  |-  ( ( ( E `  ( F `  A )
)  =  X  /\  ( E `  ( F `
 B ) )  =  Y )  -> 
( ( E `  ( F `  A ) )  =  ( E `
 ( F `  B ) )  <->  X  =  Y ) )
1817notbid 294 . . . . . 6  |-  ( ( ( E `  ( F `  A )
)  =  X  /\  ( E `  ( F `
 B ) )  =  Y )  -> 
( -.  ( E `
 ( F `  A ) )  =  ( E `  ( F `  B )
)  <->  -.  X  =  Y ) )
19 df-ne 2664 . . . . . . 7  |-  ( X  =/=  Y  <->  -.  X  =  Y )
2019biimpri 206 . . . . . 6  |-  ( -.  X  =  Y  ->  X  =/=  Y )
2118, 20syl6bi 228 . . . . 5  |-  ( ( ( E `  ( F `  A )
)  =  X  /\  ( E `  ( F `
 B ) )  =  Y )  -> 
( -.  ( E `
 ( F `  A ) )  =  ( E `  ( F `  B )
)  ->  X  =/=  Y ) )
2216, 21syl5com 30 . . . 4  |-  ( ( ( ( E : D -1-1-> R  /\  F : C -1-1-> D )  /\  ( A  e.  C  /\  B  e.  C
) )  /\  -.  ( F `  A )  =  ( F `  B ) )  -> 
( ( ( E `
 ( F `  A ) )  =  X  /\  ( E `
 ( F `  B ) )  =  Y )  ->  X  =/=  Y ) )
2322ex 434 . . 3  |-  ( ( ( E : D -1-1-> R  /\  F : C -1-1-> D )  /\  ( A  e.  C  /\  B  e.  C ) )  -> 
( -.  ( F `
 A )  =  ( F `  B
)  ->  ( (
( E `  ( F `  A )
)  =  X  /\  ( E `  ( F `
 B ) )  =  Y )  ->  X  =/=  Y ) ) )
24233adant3 1016 . 2  |-  ( ( ( E : D -1-1-> R  /\  F : C -1-1-> D )  /\  ( A  e.  C  /\  B  e.  C )  /\  A  =/=  B )  ->  ( -.  ( F `  A
)  =  ( F `
 B )  -> 
( ( ( E `
 ( F `  A ) )  =  X  /\  ( E `
 ( F `  B ) )  =  Y )  ->  X  =/=  Y ) ) )
254, 24mpd 15 1  |-  ( ( ( E : D -1-1-> R  /\  F : C -1-1-> D )  /\  ( A  e.  C  /\  B  e.  C )  /\  A  =/=  B )  ->  (
( ( E `  ( F `  A ) )  =  X  /\  ( E `  ( F `
 B ) )  =  Y )  ->  X  =/=  Y ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   -->wf 5590   -1-1->wf1 5591   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fv 5602
This theorem is referenced by:  usgra2pthspth  32141  usgra2pthlem1  32143
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