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Theorem fnexALT 5925
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 5493. This version of fnex 5924 uses ax-pow 4341, whereas fnex 5924 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 5506 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
2 relssdmrn 5353 . . . 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 5507 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
65eleq1d 2474 . . . 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 5505 . . . . 5  |-  ( F  Fn  A  ->  Fun  F )
9 funimaexg 5493 . . . . 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 5178 . . . . . . 7  |-  ( F
" dom  F )  =  ran  F
125imaeq2d 5166 . . . . . . 7  |-  ( F  Fn  A  ->  ( F " dom  F )  =  ( F " A ) )
1311, 12syl5eqr 2454 . . . . . 6  |-  ( F  Fn  A  ->  ran  F  =  ( F " A ) )
1413eleq1d 2474 . . . . 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 4952 . . 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 4313 . 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 1721   _Vcvv 2920    C_ wss 3284    X. cxp 4839   dom cdm 4841   ran crn 4842   "cima 4844   Rel wrel 4846   Fun wfun 5411    Fn wfn 5412
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-fun 5419  df-fn 5420
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