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Theorem fnct 27194
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 27189 . . . . 5  |-  ( A  ~<_  om  ->  A  e.  _V )
21adantl 466 . . . 4  |-  ( ( F  Fn  A  /\  A  ~<_  om )  ->  A  e.  _V )
3 fndm 5671 . . . . . . . 8  |-  ( F  Fn  A  ->  dom  F  =  A )
43eleq1d 2529 . . . . . . 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 5669 . . . . . 6  |-  ( F  Fn  A  ->  Fun  F )
87adantr 465 . . . . 5  |-  ( ( F  Fn  A  /\  A  ~<_  om )  ->  Fun  F )
9 funrnex 6741 . . . . 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 6702 . . . 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 5729 . . . . 5  |-  ( F  Fn  A  <->  F : A
--> ran  F )
1513, 14sylib 196 . . . 4  |-  ( ( F  Fn  A  /\  A  ~<_  om )  ->  F : A --> ran  F )
16 fssxp 5734 . . . 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 7551 . . 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 7604 . . . . 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 8049 . . . . 5  |-  om  e.  _V
23 fnrndomg 8902 . . . . . . 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 7558 . . . . . 6  |-  ( ( ran  F  ~<_  A  /\  A  ~<_  om )  ->  ran  F  ~<_  om )
2624, 25sylancom 667 . . . . 5  |-  ( ( F  Fn  A  /\  A  ~<_  om )  ->  ran  F  ~<_  om )
27 xpdom2g 7603 . . . . 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 7558 . . . 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 8382 . . 3  |-  ( om 
X.  om )  ~~  om
32 domentr 7564 . . 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 7558 . 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 1762   _Vcvv 3106    C_ wss 3469   class class class wbr 4440    X. cxp 4990   dom cdm 4992   ran crn 4993   Fun wfun 5573    Fn wfn 5574   -->wf 5575   omcom 6671    ~~ cen 7503    ~<_ cdom 7504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-ac2 8832
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-oi 7924  df-card 8309  df-acn 8312  df-ac 8486
This theorem is referenced by:  mptct  27199  mpt2cti  27200  mptctf  27202
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