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

Theorem fnexALT 5742
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 5329. This version of fnex 5741 uses ax-pow 4188, whereas fnex 5741 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 5342 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
2 relssdmrn 5193 . . . 4  |-  ( Rel 
F  ->  F  C_  ( dom  F  X.  ran  F
) )
31, 2syl 15 . . 3  |-  ( F  Fn  A  ->  F  C_  ( dom  F  X.  ran  F ) )
43adantr 451 . 2  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  C_  ( dom  F  X.  ran  F ) )
5 fndm 5343 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
65eleq1d 2349 . . . 4  |-  ( F  Fn  A  ->  ( dom  F  e.  B  <->  A  e.  B ) )
76biimpar 471 . . 3  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  dom  F  e.  B
)
8 fnfun 5341 . . . . 5  |-  ( F  Fn  A  ->  Fun  F )
9 funimaexg 5329 . . . . 5  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F " A )  e. 
_V )
108, 9sylan 457 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  ( F " A
)  e.  _V )
11 imadmrn 5024 . . . . . . 7  |-  ( F
" dom  F )  =  ran  F
125imaeq2d 5012 . . . . . . 7  |-  ( F  Fn  A  ->  ( F " dom  F )  =  ( F " A ) )
1311, 12syl5eqr 2329 . . . . . 6  |-  ( F  Fn  A  ->  ran  F  =  ( F " A ) )
1413eleq1d 2349 . . . . 5  |-  ( F  Fn  A  ->  ( ran  F  e.  _V  <->  ( F " A )  e.  _V ) )
1514biimpar 471 . . . 4  |-  ( ( F  Fn  A  /\  ( F " A )  e.  _V )  ->  ran  F  e.  _V )
1610, 15syldan 456 . . 3  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  ran  F  e.  _V )
17 xpexg 4800 . . 3  |-  ( ( dom  F  e.  B  /\  ran  F  e.  _V )  ->  ( dom  F  X.  ran  F )  e. 
_V )
187, 16, 17syl2anc 642 . 2  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  ( dom  F  X.  ran  F )  e.  _V )
19 ssexg 4160 . 2  |-  ( ( F  C_  ( dom  F  X.  ran  F )  /\  ( dom  F  X.  ran  F )  e. 
_V )  ->  F  e.  _V )
204, 18, 19syl2anc 642 1  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   _Vcvv 2788    C_ wss 3152    X. cxp 4687   dom cdm 4689   ran crn 4690   "cima 4692   Rel wrel 4694   Fun wfun 5249    Fn wfn 5250
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258
  Copyright terms: Public domain W3C validator