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Theorem fnexALT 5676
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 5267. (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 5280 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
2 relssdmrn 5180 . . . 4  |-  ( Rel 
F  ->  F  C_  ( dom  F  X.  ran  F
) )
31, 2syl 17 . . 3  |-  ( F  Fn  A  ->  F  C_  ( dom  F  X.  ran  F ) )
43adantr 453 . 2  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  C_  ( dom  F  X.  ran  F ) )
5 fndm 5281 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
65eleq1d 2324 . . . 4  |-  ( F  Fn  A  ->  ( dom  F  e.  B  <->  A  e.  B ) )
76biimpar 473 . . 3  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  dom  F  e.  B
)
8 fnfun 5279 . . . . 5  |-  ( F  Fn  A  ->  Fun  F )
9 funimaexg 5267 . . . . 5  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F " A )  e. 
_V )
108, 9sylan 459 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  ( F " A
)  e.  _V )
11 imadmrn 5012 . . . . . . 7  |-  ( F
" dom  F )  =  ran  F
125imaeq2d 5000 . . . . . . 7  |-  ( F  Fn  A  ->  ( F " dom  F )  =  ( F " A ) )
1311, 12syl5eqr 2304 . . . . . 6  |-  ( F  Fn  A  ->  ran  F  =  ( F " A ) )
1413eleq1d 2324 . . . . 5  |-  ( F  Fn  A  ->  ( ran  F  e.  _V  <->  ( F " A )  e.  _V ) )
1514biimpar 473 . . . 4  |-  ( ( F  Fn  A  /\  ( F " A )  e.  _V )  ->  ran  F  e.  _V )
1610, 15syldan 458 . . 3  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  ran  F  e.  _V )
17 xpexg 4788 . . 3  |-  ( ( dom  F  e.  B  /\  ran  F  e.  _V )  ->  ( dom  F  X.  ran  F )  e. 
_V )
187, 16, 17syl2anc 645 . 2  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  ( dom  F  X.  ran  F )  e.  _V )
19 ssexg 4134 . 2  |-  ( ( F  C_  ( dom  F  X.  ran  F )  /\  ( dom  F  X.  ran  F )  e. 
_V )  ->  F  e.  _V )
204, 18, 19syl2anc 645 1  |-  ( ( F  Fn  A  /\  A  e.  B )  ->  F  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    e. wcel 1621   _Vcvv 2763    C_ wss 3127    X. cxp 4659   dom cdm 4661   ran crn 4662   "cima 4664   Rel wrel 4666   Fun wfun 4667    Fn wfn 4668
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684
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