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

Theorem fnexALT 5962
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 5530. This version of fnex 5961 uses ax-pow 4377, whereas fnex 5961 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 5543 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
2 relssdmrn 5390 . . . 4  |-  ( Rel 
F  ->  F  C_  ( dom  F  X.  ran  F
) )
31, 2syl 16 . . 3  |-  ( F  Fn  A  ->  F  C_  ( dom  F  X.  ran  F ) )
43adantr 452 . 2  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  C_  ( dom  F  X.  ran  F ) )
5 fndm 5544 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
65eleq1d 2502 . . . 4  |-  ( F  Fn  A  ->  ( dom  F  e.  B  <->  A  e.  B ) )
76biimpar 472 . . 3  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  dom  F  e.  B
)
8 fnfun 5542 . . . . 5  |-  ( F  Fn  A  ->  Fun  F )
9 funimaexg 5530 . . . . 5  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F " A )  e. 
_V )
108, 9sylan 458 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  ( F " A
)  e.  _V )
11 imadmrn 5215 . . . . . . 7  |-  ( F
" dom  F )  =  ran  F
125imaeq2d 5203 . . . . . . 7  |-  ( F  Fn  A  ->  ( F " dom  F )  =  ( F " A ) )
1311, 12syl5eqr 2482 . . . . . 6  |-  ( F  Fn  A  ->  ran  F  =  ( F " A ) )
1413eleq1d 2502 . . . . 5  |-  ( F  Fn  A  ->  ( ran  F  e.  _V  <->  ( F " A )  e.  _V ) )
1514biimpar 472 . . . 4  |-  ( ( F  Fn  A  /\  ( F " A )  e.  _V )  ->  ran  F  e.  _V )
1610, 15syldan 457 . . 3  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  ran  F  e.  _V )
17 xpexg 4989 . . 3  |-  ( ( dom  F  e.  B  /\  ran  F  e.  _V )  ->  ( dom  F  X.  ran  F )  e. 
_V )
187, 16, 17syl2anc 643 . 2  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  ( dom  F  X.  ran  F )  e.  _V )
19 ssexg 4349 . 2  |-  ( ( F  C_  ( dom  F  X.  ran  F )  /\  ( dom  F  X.  ran  F )  e. 
_V )  ->  F  e.  _V )
204, 18, 19syl2anc 643 1  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   _Vcvv 2956    C_ wss 3320    X. cxp 4876   dom cdm 4878   ran crn 4879   "cima 4881   Rel wrel 4883   Fun wfun 5448    Fn wfn 5449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-fun 5456  df-fn 5457
  Copyright terms: Public domain W3C validator