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Theorem fnct 27514
Description: If the domain of a function is countable, the function is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
Assertion
Ref Expression
fnct  |-  ( ( F  Fn  A  /\  A  ~<_  om )  ->  F  ~<_  om )

Proof of Theorem fnct
StepHypRef Expression
1 ctex 27509 . . . . 5  |-  ( A  ~<_  om  ->  A  e.  _V )
21adantl 466 . . . 4  |-  ( ( F  Fn  A  /\  A  ~<_  om )  ->  A  e.  _V )
3 fndm 5670 . . . . . . . 8  |-  ( F  Fn  A  ->  dom  F  =  A )
43eleq1d 2512 . . . . . . 7  |-  ( F  Fn  A  ->  ( dom  F  e.  _V  <->  A  e.  _V ) )
54adantr 465 . . . . . 6  |-  ( ( F  Fn  A  /\  A  ~<_  om )  ->  ( dom  F  e.  _V  <->  A  e.  _V ) )
62, 5mpbird 232 . . . . 5  |-  ( ( F  Fn  A  /\  A  ~<_  om )  ->  dom  F  e.  _V )
7 fnfun 5668 . . . . . 6  |-  ( F  Fn  A  ->  Fun  F )
87adantr 465 . . . . 5  |-  ( ( F  Fn  A  /\  A  ~<_  om )  ->  Fun  F )
9 funrnex 6752 . . . . 5  |-  ( dom 
F  e.  _V  ->  ( Fun  F  ->  ran  F  e.  _V ) )
106, 8, 9sylc 60 . . . 4  |-  ( ( F  Fn  A  /\  A  ~<_  om )  ->  ran  F  e.  _V )
11 xpexg 6587 . . . 4  |-  ( ( A  e.  _V  /\  ran  F  e.  _V )  ->  ( A  X.  ran  F )  e.  _V )
122, 10, 11syl2anc 661 . . 3  |-  ( ( F  Fn  A  /\  A  ~<_  om )  ->  ( A  X.  ran  F )  e.  _V )
13 simpl 457 . . . . 5  |-  ( ( F  Fn  A  /\  A  ~<_  om )  ->  F  Fn  A )
14 dffn3 5728 . . . . 5  |-  ( F  Fn  A  <->  F : A
--> ran  F )
1513, 14sylib 196 . . . 4  |-  ( ( F  Fn  A  /\  A  ~<_  om )  ->  F : A --> ran  F )
16 fssxp 5733 . . . 4  |-  ( F : A --> ran  F  ->  F  C_  ( A  X.  ran  F ) )
1715, 16syl 16 . . 3  |-  ( ( F  Fn  A  /\  A  ~<_  om )  ->  F  C_  ( A  X.  ran  F ) )
18 ssdomg 7563 . . 3  |-  ( ( A  X.  ran  F
)  e.  _V  ->  ( F  C_  ( A  X.  ran  F )  ->  F  ~<_  ( A  X.  ran  F ) ) )
1912, 17, 18sylc 60 . 2  |-  ( ( F  Fn  A  /\  A  ~<_  om )  ->  F  ~<_  ( A  X.  ran  F
) )
20 xpdom1g 7616 . . . . 5  |-  ( ( ran  F  e.  _V  /\  A  ~<_  om )  ->  ( A  X.  ran  F )  ~<_  ( om  X.  ran  F ) )
2110, 20sylancom 667 . . . 4  |-  ( ( F  Fn  A  /\  A  ~<_  om )  ->  ( A  X.  ran  F )  ~<_  ( om  X.  ran  F ) )
22 omex 8063 . . . . 5  |-  om  e.  _V
23 fnrndomg 8916 . . . . . . 7  |-  ( A  e.  _V  ->  ( F  Fn  A  ->  ran 
F  ~<_  A ) )
242, 13, 23sylc 60 . . . . . 6  |-  ( ( F  Fn  A  /\  A  ~<_  om )  ->  ran  F  ~<_  A )
25 domtr 7570 . . . . . 6  |-  ( ( ran  F  ~<_  A  /\  A  ~<_  om )  ->  ran  F  ~<_  om )
2624, 25sylancom 667 . . . . 5  |-  ( ( F  Fn  A  /\  A  ~<_  om )  ->  ran  F  ~<_  om )
27 xpdom2g 7615 . . . . 5  |-  ( ( om  e.  _V  /\  ran  F  ~<_  om )  ->  ( om  X.  ran  F )  ~<_  ( om  X.  om ) )
2822, 26, 27sylancr 663 . . . 4  |-  ( ( F  Fn  A  /\  A  ~<_  om )  ->  ( om  X.  ran  F )  ~<_  ( om  X.  om ) )
29 domtr 7570 . . . 4  |-  ( ( ( A  X.  ran  F )  ~<_  ( om  X.  ran  F )  /\  ( om  X.  ran  F )  ~<_  ( om  X.  om ) )  ->  ( A  X.  ran  F )  ~<_  ( om  X.  om ) )
3021, 28, 29syl2anc 661 . . 3  |-  ( ( F  Fn  A  /\  A  ~<_  om )  ->  ( A  X.  ran  F )  ~<_  ( om  X.  om ) )
31 xpomen 8396 . . 3  |-  ( om 
X.  om )  ~~  om
32 domentr 7576 . . 3  |-  ( ( ( A  X.  ran  F )  ~<_  ( om  X.  om )  /\  ( om  X.  om )  ~~  om )  ->  ( A  X.  ran  F )  ~<_  om )
3330, 31, 32sylancl 662 . 2  |-  ( ( F  Fn  A  /\  A  ~<_  om )  ->  ( A  X.  ran  F )  ~<_  om )
34 domtr 7570 . 2  |-  ( ( F  ~<_  ( A  X.  ran  F )  /\  ( A  X.  ran  F )  ~<_  om )  ->  F  ~<_  om )
3519, 33, 34syl2anc 661 1  |-  ( ( F  Fn  A  /\  A  ~<_  om )  ->  F  ~<_  om )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1804   _Vcvv 3095    C_ wss 3461   class class class wbr 4437    X. cxp 4987   dom cdm 4989   ran crn 4990   Fun wfun 5572    Fn wfn 5573   -->wf 5574   omcom 6685    ~~ cen 7515    ~<_ cdom 7516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-ac2 8846
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-oi 7938  df-card 8323  df-acn 8326  df-ac 8500
This theorem is referenced by:  mptct  27519  mpt2cti  27520  mptctf  27522
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