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Theorem 2f1fvneq 39158
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 6179 . . . . 5  |-  ( ( F : C -1-1-> D  /\  ( A  e.  C  /\  B  e.  C
) )  ->  (
( F `  A
)  =  ( F `
 B )  ->  A  =  B )
)
21adantll 728 . . . 4  |-  ( ( ( E : D -1-1-> R  /\  F : C -1-1-> D )  /\  ( A  e.  C  /\  B  e.  C ) )  -> 
( ( F `  A )  =  ( F `  B )  ->  A  =  B ) )
32necon3ad 2656 . . 3  |-  ( ( ( E : D -1-1-> R  /\  F : C -1-1-> D )  /\  ( A  e.  C  /\  B  e.  C ) )  -> 
( A  =/=  B  ->  -.  ( F `  A )  =  ( F `  B ) ) )
433impia 1228 . 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 768 . . . . . . 7  |-  ( ( ( E : D -1-1-> R  /\  F : C -1-1-> D )  /\  ( A  e.  C  /\  B  e.  C ) )  ->  E : D -1-1-> R )
6 f1f 5792 . . . . . . . . . 10  |-  ( F : C -1-1-> D  ->  F : C --> D )
7 ffvelrn 6035 . . . . . . . . . . . 12  |-  ( ( F : C --> D  /\  A  e.  C )  ->  ( F `  A
)  e.  D )
8 ffvelrn 6035 . . . . . . . . . . . 12  |-  ( ( F : C --> D  /\  B  e.  C )  ->  ( F `  B
)  e.  D )
97, 8anim12dan 855 . . . . . . . . . . 11  |-  ( ( F : C --> D  /\  ( A  e.  C  /\  B  e.  C
) )  ->  (
( F `  A
)  e.  D  /\  ( F `  B )  e.  D ) )
109ex 441 . . . . . . . . . 10  |-  ( F : C --> D  -> 
( ( A  e.  C  /\  B  e.  C )  ->  (
( F `  A
)  e.  D  /\  ( F `  B )  e.  D ) ) )
116, 10syl 17 . . . . . . . . 9  |-  ( F : C -1-1-> D  -> 
( ( A  e.  C  /\  B  e.  C )  ->  (
( F `  A
)  e.  D  /\  ( F `  B )  e.  D ) ) )
1211adantl 473 . . . . . . . 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 436 . . . . . . 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 6179 . . . . . . 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 673 . . . . . 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 448 . . . . 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 2484 . . . . . . 7  |-  ( ( ( E `  ( F `  A )
)  =  X  /\  ( E `  ( F `
 B ) )  =  Y )  -> 
( ( E `  ( F `  A ) )  =  ( E `
 ( F `  B ) )  <->  X  =  Y ) )
1817notbid 301 . . . . . 6  |-  ( ( ( E `  ( F `  A )
)  =  X  /\  ( E `  ( F `
 B ) )  =  Y )  -> 
( -.  ( E `
 ( F `  A ) )  =  ( E `  ( F `  B )
)  <->  -.  X  =  Y ) )
19 df-ne 2643 . . . . . . 7  |-  ( X  =/=  Y  <->  -.  X  =  Y )
2019biimpri 211 . . . . . 6  |-  ( -.  X  =  Y  ->  X  =/=  Y )
2118, 20syl6bi 236 . . . . 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 441 . . 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 1050 . 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 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   -->wf 5585   -1-1->wf1 5586   ` cfv 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fv 5597
This theorem is referenced by:  usgra2pthspth  40173  usgra2pthlem1  40175
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