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Theorem unirnfdomd 8992
Description: The union of the range of a function from an infinite set into the class of finite sets is dominated by its domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
unirnfdomd.1  |-  ( ph  ->  F : T --> Fin )
unirnfdomd.2  |-  ( ph  ->  -.  T  e.  Fin )
unirnfdomd.3  |-  ( ph  ->  T  e.  V )
Assertion
Ref Expression
unirnfdomd  |-  ( ph  ->  U. ran  F  ~<_  T )

Proof of Theorem unirnfdomd
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 unirnfdomd.1 . . . . . . . 8  |-  ( ph  ->  F : T --> Fin )
2 ffn 5728 . . . . . . . 8  |-  ( F : T --> Fin  ->  F  Fn  T )
31, 2syl 17 . . . . . . 7  |-  ( ph  ->  F  Fn  T )
4 unirnfdomd.3 . . . . . . 7  |-  ( ph  ->  T  e.  V )
5 fnex 6132 . . . . . . 7  |-  ( ( F  Fn  T  /\  T  e.  V )  ->  F  e.  _V )
63, 4, 5syl2anc 667 . . . . . 6  |-  ( ph  ->  F  e.  _V )
7 rnexg 6725 . . . . . 6  |-  ( F  e.  _V  ->  ran  F  e.  _V )
86, 7syl 17 . . . . 5  |-  ( ph  ->  ran  F  e.  _V )
9 frn 5735 . . . . . . 7  |-  ( F : T --> Fin  ->  ran 
F  C_  Fin )
10 dfss3 3422 . . . . . . 7  |-  ( ran 
F  C_  Fin  <->  A. x  e.  ran  F  x  e. 
Fin )
119, 10sylib 200 . . . . . 6  |-  ( F : T --> Fin  ->  A. x  e.  ran  F  x  e.  Fin )
12 isfinite 8157 . . . . . . . 8  |-  ( x  e.  Fin  <->  x  ~<  om )
13 sdomdom 7597 . . . . . . . 8  |-  ( x 
~<  om  ->  x  ~<_  om )
1412, 13sylbi 199 . . . . . . 7  |-  ( x  e.  Fin  ->  x  ~<_  om )
1514ralimi 2781 . . . . . 6  |-  ( A. x  e.  ran  F  x  e.  Fin  ->  A. x  e.  ran  F  x  ~<_  om )
161, 11, 153syl 18 . . . . 5  |-  ( ph  ->  A. x  e.  ran  F  x  ~<_  om )
17 unidom 8968 . . . . 5  |-  ( ( ran  F  e.  _V  /\ 
A. x  e.  ran  F  x  ~<_  om )  ->  U. ran  F  ~<_  ( ran  F  X.  om ) )
188, 16, 17syl2anc 667 . . . 4  |-  ( ph  ->  U. ran  F  ~<_  ( ran  F  X.  om ) )
19 fnrndomg 8963 . . . . . 6  |-  ( T  e.  V  ->  ( F  Fn  T  ->  ran 
F  ~<_  T ) )
204, 3, 19sylc 62 . . . . 5  |-  ( ph  ->  ran  F  ~<_  T )
21 omex 8148 . . . . . 6  |-  om  e.  _V
2221xpdom1 7671 . . . . 5  |-  ( ran 
F  ~<_  T  ->  ( ran  F  X.  om )  ~<_  ( T  X.  om )
)
2320, 22syl 17 . . . 4  |-  ( ph  ->  ( ran  F  X.  om )  ~<_  ( T  X.  om ) )
24 domtr 7622 . . . 4  |-  ( ( U. ran  F  ~<_  ( ran  F  X.  om )  /\  ( ran  F  X.  om )  ~<_  ( T  X.  om ) )  ->  U. ran  F  ~<_  ( T  X.  om )
)
2518, 23, 24syl2anc 667 . . 3  |-  ( ph  ->  U. ran  F  ~<_  ( T  X.  om )
)
26 unirnfdomd.2 . . . . 5  |-  ( ph  ->  -.  T  e.  Fin )
27 infinf 8991 . . . . . 6  |-  ( T  e.  V  ->  ( -.  T  e.  Fin  <->  om  ~<_  T ) )
284, 27syl 17 . . . . 5  |-  ( ph  ->  ( -.  T  e. 
Fin 
<->  om  ~<_  T ) )
2926, 28mpbid 214 . . . 4  |-  ( ph  ->  om  ~<_  T )
30 xpdom2g 7668 . . . 4  |-  ( ( T  e.  V  /\  om  ~<_  T )  ->  ( T  X.  om )  ~<_  ( T  X.  T ) )
314, 29, 30syl2anc 667 . . 3  |-  ( ph  ->  ( T  X.  om )  ~<_  ( T  X.  T ) )
32 domtr 7622 . . 3  |-  ( ( U. ran  F  ~<_  ( T  X.  om )  /\  ( T  X.  om )  ~<_  ( T  X.  T ) )  ->  U. ran  F  ~<_  ( T  X.  T ) )
3325, 31, 32syl2anc 667 . 2  |-  ( ph  ->  U. ran  F  ~<_  ( T  X.  T ) )
34 infxpidm 8987 . . 3  |-  ( om  ~<_  T  ->  ( T  X.  T )  ~~  T
)
3529, 34syl 17 . 2  |-  ( ph  ->  ( T  X.  T
)  ~~  T )
36 domentr 7628 . 2  |-  ( ( U. ran  F  ~<_  ( T  X.  T )  /\  ( T  X.  T )  ~~  T
)  ->  U. ran  F  ~<_  T )
3733, 35, 36syl2anc 667 1  |-  ( ph  ->  U. ran  F  ~<_  T )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    e. wcel 1887   A.wral 2737   _Vcvv 3045    C_ wss 3404   U.cuni 4198   class class class wbr 4402    X. cxp 4832   ran crn 4835    Fn wfn 5577   -->wf 5578   omcom 6692    ~~ cen 7566    ~<_ cdom 7567    ~< csdm 7568   Fincfn 7569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-ac2 8893
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-oi 8025  df-card 8373  df-acn 8376  df-ac 8547
This theorem is referenced by:  acsdomd  16427
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