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Theorem unirnfdomd 8189
Description: The union of the range of a function from a 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 5389 . . . . . . . 8  |-  ( F : T --> Fin  ->  F  Fn  T )
31, 2syl 15 . . . . . . 7  |-  ( ph  ->  F  Fn  T )
4 unirnfdomd.3 . . . . . . 7  |-  ( ph  ->  T  e.  V )
5 fnex 5741 . . . . . . 7  |-  ( ( F  Fn  T  /\  T  e.  V )  ->  F  e.  _V )
63, 4, 5syl2anc 642 . . . . . 6  |-  ( ph  ->  F  e.  _V )
7 rnexg 4940 . . . . . 6  |-  ( F  e.  _V  ->  ran  F  e.  _V )
86, 7syl 15 . . . . 5  |-  ( ph  ->  ran  F  e.  _V )
9 frn 5395 . . . . . . 7  |-  ( F : T --> Fin  ->  ran 
F  C_  Fin )
10 dfss3 3170 . . . . . . 7  |-  ( ran 
F  C_  Fin  <->  A. x  e.  ran  F  x  e. 
Fin )
119, 10sylib 188 . . . . . 6  |-  ( F : T --> Fin  ->  A. x  e.  ran  F  x  e.  Fin )
12 isfinite 7353 . . . . . . . 8  |-  ( x  e.  Fin  <->  x  ~<  om )
13 sdomdom 6889 . . . . . . . 8  |-  ( x 
~<  om  ->  x  ~<_  om )
1412, 13sylbi 187 . . . . . . 7  |-  ( x  e.  Fin  ->  x  ~<_  om )
1514ralimi 2618 . . . . . 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 8165 . . . . 5  |-  ( ( ran  F  e.  _V  /\ 
A. x  e.  ran  F  x  ~<_  om )  ->  U. ran  F  ~<_  ( ran  F  X.  om ) )
188, 16, 17syl2anc 642 . . . 4  |-  ( ph  ->  U. ran  F  ~<_  ( ran  F  X.  om ) )
19 fnrndomg 8160 . . . . . 6  |-  ( T  e.  V  ->  ( F  Fn  T  ->  ran 
F  ~<_  T ) )
204, 3, 19sylc 56 . . . . 5  |-  ( ph  ->  ran  F  ~<_  T )
21 omex 7344 . . . . . 6  |-  om  e.  _V
2221xpdom1 6961 . . . . 5  |-  ( ran 
F  ~<_  T  ->  ( ran  F  X.  om )  ~<_  ( T  X.  om )
)
2320, 22syl 15 . . . 4  |-  ( ph  ->  ( ran  F  X.  om )  ~<_  ( T  X.  om ) )
24 domtr 6914 . . . 4  |-  ( ( U. ran  F  ~<_  ( ran  F  X.  om )  /\  ( ran  F  X.  om )  ~<_  ( T  X.  om ) )  ->  U. ran  F  ~<_  ( T  X.  om )
)
2518, 23, 24syl2anc 642 . . 3  |-  ( ph  ->  U. ran  F  ~<_  ( T  X.  om )
)
26 unirnfdomd.2 . . . . 5  |-  ( ph  ->  -.  T  e.  Fin )
27 infinf 8188 . . . . . 6  |-  ( T  e.  V  ->  ( -.  T  e.  Fin  <->  om  ~<_  T ) )
284, 27syl 15 . . . . 5  |-  ( ph  ->  ( -.  T  e. 
Fin 
<->  om  ~<_  T ) )
2926, 28mpbid 201 . . . 4  |-  ( ph  ->  om  ~<_  T )
30 xpdom2g 6958 . . . 4  |-  ( ( T  e.  V  /\  om  ~<_  T )  ->  ( T  X.  om )  ~<_  ( T  X.  T ) )
314, 29, 30syl2anc 642 . . 3  |-  ( ph  ->  ( T  X.  om )  ~<_  ( T  X.  T ) )
32 domtr 6914 . . 3  |-  ( ( U. ran  F  ~<_  ( T  X.  om )  /\  ( T  X.  om )  ~<_  ( T  X.  T ) )  ->  U. ran  F  ~<_  ( T  X.  T ) )
3325, 31, 32syl2anc 642 . 2  |-  ( ph  ->  U. ran  F  ~<_  ( T  X.  T ) )
34 infxpidm 8184 . . 3  |-  ( om  ~<_  T  ->  ( T  X.  T )  ~~  T
)
3529, 34syl 15 . 2  |-  ( ph  ->  ( T  X.  T
)  ~~  T )
36 domentr 6920 . 2  |-  ( ( U. ran  F  ~<_  ( T  X.  T )  /\  ( T  X.  T )  ~~  T
)  ->  U. ran  F  ~<_  T )
3733, 35, 36syl2anc 642 1  |-  ( ph  ->  U. ran  F  ~<_  T )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   U.cuni 3827   class class class wbr 4023   omcom 4656    X. cxp 4687   ran crn 4690    Fn wfn 5250   -->wf 5251    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862   Fincfn 6863
This theorem is referenced by:  acsdomd  14284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-ac2 8089
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-acn 7575  df-ac 7743
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