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

Theorem fimacnv 5673
Description: The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)
Assertion
Ref Expression
fimacnv  |-  ( F : A --> B  -> 
( `' F " B )  =  A )

Proof of Theorem fimacnv
StepHypRef Expression
1 imassrn 5041 . . 3  |-  ( `' F " B ) 
C_  ran  `' F
2 dfdm4 4888 . . . 4  |-  dom  F  =  ran  `' F
3 fdm 5409 . . . . 5  |-  ( F : A --> B  ->  dom  F  =  A )
4 ssid 3210 . . . . . 6  |-  A  C_  A
54a1i 10 . . . . 5  |-  ( F : A --> B  ->  A  C_  A )
63, 5eqsstrd 3225 . . . 4  |-  ( F : A --> B  ->  dom  F  C_  A )
72, 6syl5eqssr 3236 . . 3  |-  ( F : A --> B  ->  ran  `' F  C_  A )
81, 7syl5ss 3203 . 2  |-  ( F : A --> B  -> 
( `' F " B )  C_  A
)
9 imassrn 5041 . . . 4  |-  ( F
" A )  C_  ran  F
10 frn 5411 . . . 4  |-  ( F : A --> B  ->  ran  F  C_  B )
119, 10syl5ss 3203 . . 3  |-  ( F : A --> B  -> 
( F " A
)  C_  B )
12 ffun 5407 . . . 4  |-  ( F : A --> B  ->  Fun  F )
134, 3syl5sseqr 3240 . . . 4  |-  ( F : A --> B  ->  A  C_  dom  F )
14 funimass3 5657 . . . 4  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A 
C_  ( `' F " B ) ) )
1512, 13, 14syl2anc 642 . . 3  |-  ( F : A --> B  -> 
( ( F " A )  C_  B  <->  A 
C_  ( `' F " B ) ) )
1611, 15mpbid 201 . 2  |-  ( F : A --> B  ->  A  C_  ( `' F " B ) )
178, 16eqssd 3209 1  |-  ( F : A --> B  -> 
( `' F " B )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    C_ wss 3165   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708   Fun wfun 5265   -->wf 5267
This theorem is referenced by:  fmpt  5697  fin1a2lem7  8048  nn0supp  10033  cnclima  17013  iscncl  17014  cnindis  17036  cncmp  17135  ptrescn  17349  qtopuni  17409  qtopcld  17420  qtopcmap  17426  ordthmeolem  17508  rnelfmlem  17663  mbfdm  18999  ismbf  19001  mbfimaicc  19004  ismbf2d  19012  ismbf3d  19025  mbfimaopn2  19028  i1fd  19052  plyeq0  19609  fimacnvinrn  23214  imambfm  23582  dstrvprob  23687  intopcoaconlem3b  25641  intopcoaconlem3  25642  fsuppeq  27362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279
  Copyright terms: Public domain W3C validator