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Theorem f1imaen 7484
Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by NM, 30-Sep-2004.)
Hypothesis
Ref Expression
f1imaen.1  |-  C  e. 
_V
Assertion
Ref Expression
f1imaen  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F " C )  ~~  C
)

Proof of Theorem f1imaen
StepHypRef Expression
1 f1imaen.1 . 2  |-  C  e. 
_V
2 f1imaeng 7482 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A  /\  C  e.  _V )  ->  ( F " C
)  ~~  C )
31, 2mp3an3 1304 1  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F " C )  ~~  C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1758   _Vcvv 3078    C_ wss 3439   class class class wbr 4403   "cima 4954   -1-1->wf1 5526    ~~ cen 7420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-er 7214  df-en 7424
This theorem is referenced by:  ssenen  7598  fin4en1  8592  tskinf  9050  tskuni  9064  isercoll  13266  phimullem  13975  odngen  16200  erdsze2lem2  27256
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