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Theorem fnexALT 5704
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 5295. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.)
Assertion
Ref Expression
fnexALT  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  e.  _V )

Proof of Theorem fnexALT
StepHypRef Expression
1 fnrel 5308 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
2 relssdmrn 5191 . . . 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 5309 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
65eleq1d 2350 . . . 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 5307 . . . . 5  |-  ( F  Fn  A  ->  Fun  F )
9 funimaexg 5295 . . . . 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 5023 . . . . . . 7  |-  ( F
"  dom  F )  =  ran  F
125imaeq2d 5011 . . . . . . 7  |-  ( F  Fn  A  ->  ( F "  dom  F )  =  ( F " A ) )
1311, 12syl5eqr 2330 . . . . . 6  |-  ( F  Fn  A  ->  ran  F  =  ( F " A ) )
1413eleq1d 2350 . . . . 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 4799 . . 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 4161 . 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 1685   _Vcvv 2789    C_ wss 3153    X. cxp 4686    dom cdm 4688   ran crn 4689   "cima 4691   Rel wrel 4693   Fun wfun 5215    Fn wfn 5216
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224
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