ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fnexALT Unicode version

Theorem fnexALT 5866
Description: If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of funimaexg 5084. This version of fnex 5501 uses ax-pow 4001 and ax-un 4251, whereas fnex 5501 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
fnexALT  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  e.  _V )

Proof of Theorem fnexALT
StepHypRef Expression
1 fnrel 5098 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
2 relssdmrn 4938 . . . 4  |-  ( Rel 
F  ->  F  C_  ( dom  F  X.  ran  F
) )
31, 2syl 14 . . 3  |-  ( F  Fn  A  ->  F  C_  ( dom  F  X.  ran  F ) )
43adantr 270 . 2  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  C_  ( dom  F  X.  ran  F ) )
5 fndm 5099 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
65eleq1d 2156 . . . 4  |-  ( F  Fn  A  ->  ( dom  F  e.  B  <->  A  e.  B ) )
76biimpar 291 . . 3  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  dom  F  e.  B
)
8 fnfun 5097 . . . . 5  |-  ( F  Fn  A  ->  Fun  F )
9 funimaexg 5084 . . . . 5  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F " A )  e. 
_V )
108, 9sylan 277 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  ( F " A
)  e.  _V )
11 imadmrn 4771 . . . . . . 7  |-  ( F
" dom  F )  =  ran  F
125imaeq2d 4761 . . . . . . 7  |-  ( F  Fn  A  ->  ( F " dom  F )  =  ( F " A ) )
1311, 12syl5eqr 2134 . . . . . 6  |-  ( F  Fn  A  ->  ran  F  =  ( F " A ) )
1413eleq1d 2156 . . . . 5  |-  ( F  Fn  A  ->  ( ran  F  e.  _V  <->  ( F " A )  e.  _V ) )
1514biimpar 291 . . . 4  |-  ( ( F  Fn  A  /\  ( F " A )  e.  _V )  ->  ran  F  e.  _V )
1610, 15syldan 276 . . 3  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  ran  F  e.  _V )
17 xpexg 4540 . . 3  |-  ( ( dom  F  e.  B  /\  ran  F  e.  _V )  ->  ( dom  F  X.  ran  F )  e. 
_V )
187, 16, 17syl2anc 403 . 2  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  ( dom  F  X.  ran  F )  e.  _V )
19 ssexg 3970 . 2  |-  ( ( F  C_  ( dom  F  X.  ran  F )  /\  ( dom  F  X.  ran  F )  e. 
_V )  ->  F  e.  _V )
204, 18, 19syl2anc 403 1  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1438   _Vcvv 2619    C_ wss 2997    X. cxp 4426   dom cdm 4428   ran crn 4429   "cima 4431   Rel wrel 4433   Fun wfun 4996    Fn wfn 4997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-fun 5004  df-fn 5005
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator