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Theorem f1veqaeq 5970
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 5969 . . 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 5689 . . . . . . . 8  |-  ( c  =  C  ->  ( F `  c )  =  ( F `  C ) )
32eqeq1d 2449 . . . . . . 7  |-  ( c  =  C  ->  (
( F `  c
)  =  ( F `
 d )  <->  ( F `  C )  =  ( F `  d ) ) )
4 eqeq1 2447 . . . . . . 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 5689 . . . . . . . 8  |-  ( d  =  D  ->  ( F `  d )  =  ( F `  D ) )
76eqeq2d 2452 . . . . . . 7  |-  ( d  =  D  ->  (
( F `  C
)  =  ( F `
 d )  <->  ( F `  C )  =  ( F `  D ) ) )
8 eqeq2 2450 . . . . . . 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 3077 . . . . 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 1369    e. wcel 1756   A.wral 2713   -->wf 5412   -1-1->wf1 5413   ` cfv 5416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fv 5424
This theorem is referenced by:  f1fveq  5973  f1ocnvfvrneq  5988  f1o2ndf1  6678  symgfvne  15891  f1rhm0to0  16826  f1otrg  23115  spthonepeq  23484  4cycl4dv  23551  mblfinlem2  28426  2f1fvneq  30140  usg2wlkeq  30286
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