MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1imaeng Structured version   Unicode version

Theorem f1imaeng 7533
Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
f1imaeng  |-  ( ( F : A -1-1-> B  /\  C  C_  A  /\  C  e.  V )  ->  ( F " C
)  ~~  C )

Proof of Theorem f1imaeng
StepHypRef Expression
1 f1ores 5769 . . 3  |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-onto-> ( F " C ) )
2 f1oeng 7492 . . . 4  |-  ( ( C  e.  V  /\  ( F  |`  C ) : C -1-1-onto-> ( F " C
) )  ->  C  ~~  ( F " C
) )
32ancoms 451 . . 3  |-  ( ( ( F  |`  C ) : C -1-1-onto-> ( F " C
)  /\  C  e.  V )  ->  C  ~~  ( F " C
) )
41, 3stoic3 1630 . 2  |-  ( ( F : A -1-1-> B  /\  C  C_  A  /\  C  e.  V )  ->  C  ~~  ( F
" C ) )
54ensymd 7524 1  |-  ( ( F : A -1-1-> B  /\  C  C_  A  /\  C  e.  V )  ->  ( F " C
)  ~~  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 974    e. wcel 1842    C_ wss 3413   class class class wbr 4394    |` cres 4944   "cima 4945   -1-1->wf1 5522   -1-1-onto->wf1o 5524    ~~ cen 7471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-er 7268  df-en 7475
This theorem is referenced by:  f1imaen  7535  ackbij1b  8571  enfin1ai  8716  isercolllem2  13544  pmtrfconj  16707  ballotlemro  28847
  Copyright terms: Public domain W3C validator