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Theorem f1veqaeq 6154
Description: If the values of a one-to-one function for two arguments are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
Assertion
Ref Expression
f1veqaeq  |-  ( ( F : A -1-1-> B  /\  ( C  e.  A  /\  D  e.  A
) )  ->  (
( F `  C
)  =  ( F `
 D )  ->  C  =  D )
)

Proof of Theorem f1veqaeq
Dummy variables  c 
d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dff13 6152 . . 3  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  A. c  e.  A  A. d  e.  A  (
( F `  c
)  =  ( F `
 d )  -> 
c  =  d ) ) )
2 fveq2 5864 . . . . . . . 8  |-  ( c  =  C  ->  ( F `  c )  =  ( F `  C ) )
32eqeq1d 2469 . . . . . . 7  |-  ( c  =  C  ->  (
( F `  c
)  =  ( F `
 d )  <->  ( F `  C )  =  ( F `  d ) ) )
4 eqeq1 2471 . . . . . . 7  |-  ( c  =  C  ->  (
c  =  d  <->  C  =  d ) )
53, 4imbi12d 320 . . . . . 6  |-  ( c  =  C  ->  (
( ( F `  c )  =  ( F `  d )  ->  c  =  d )  <->  ( ( F `
 C )  =  ( F `  d
)  ->  C  =  d ) ) )
6 fveq2 5864 . . . . . . . 8  |-  ( d  =  D  ->  ( F `  d )  =  ( F `  D ) )
76eqeq2d 2481 . . . . . . 7  |-  ( d  =  D  ->  (
( F `  C
)  =  ( F `
 d )  <->  ( F `  C )  =  ( F `  D ) ) )
8 eqeq2 2482 . . . . . . 7  |-  ( d  =  D  ->  ( C  =  d  <->  C  =  D ) )
97, 8imbi12d 320 . . . . . 6  |-  ( d  =  D  ->  (
( ( F `  C )  =  ( F `  d )  ->  C  =  d )  <->  ( ( F `
 C )  =  ( F `  D
)  ->  C  =  D ) ) )
105, 9rspc2v 3223 . . . . 5  |-  ( ( C  e.  A  /\  D  e.  A )  ->  ( A. c  e.  A  A. d  e.  A  ( ( F `
 c )  =  ( F `  d
)  ->  c  =  d )  ->  (
( F `  C
)  =  ( F `
 D )  ->  C  =  D )
) )
1110com12 31 . . . 4  |-  ( A. c  e.  A  A. d  e.  A  (
( F `  c
)  =  ( F `
 d )  -> 
c  =  d )  ->  ( ( C  e.  A  /\  D  e.  A )  ->  (
( F `  C
)  =  ( F `
 D )  ->  C  =  D )
) )
1211adantl 466 . . 3  |-  ( ( F : A --> B  /\  A. c  e.  A  A. d  e.  A  (
( F `  c
)  =  ( F `
 d )  -> 
c  =  d ) )  ->  ( ( C  e.  A  /\  D  e.  A )  ->  ( ( F `  C )  =  ( F `  D )  ->  C  =  D ) ) )
131, 12sylbi 195 . 2  |-  ( F : A -1-1-> B  -> 
( ( C  e.  A  /\  D  e.  A )  ->  (
( F `  C
)  =  ( F `
 D )  ->  C  =  D )
) )
1413imp 429 1  |-  ( ( F : A -1-1-> B  /\  ( C  e.  A  /\  D  e.  A
) )  ->  (
( F `  C
)  =  ( F `
 D )  ->  C  =  D )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   -->wf 5582   -1-1->wf1 5583   ` cfv 5586
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 4568  ax-nul 4576  ax-pr 4686
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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fv 5594
This theorem is referenced by:  f1fveq  6156  f1ocnvfvrneq  6175  f1o2ndf1  6888  symgfvne  16205  f1rhm0to0  17169  mat2pmatf1  18994  f1otrg  23847  spthonepeq  24262  4cycl4dv  24340  usg2wlkeq  24381  mblfinlem2  29627  2f1fvneq  31776
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