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Theorem unirnfdomd 8205
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 5405 . . . . . . . 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 5757 . . . . . . 7  |-  ( ( F  Fn  T  /\  T  e.  V )  ->  F  e.  _V )
63, 4, 5syl2anc 642 . . . . . 6  |-  ( ph  ->  F  e.  _V )
7 rnexg 4956 . . . . . 6  |-  ( F  e.  _V  ->  ran  F  e.  _V )
86, 7syl 15 . . . . 5  |-  ( ph  ->  ran  F  e.  _V )
9 frn 5411 . . . . . . 7  |-  ( F : T --> Fin  ->  ran 
F  C_  Fin )
10 dfss3 3183 . . . . . . 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 7369 . . . . . . . 8  |-  ( x  e.  Fin  <->  x  ~<  om )
13 sdomdom 6905 . . . . . . . 8  |-  ( x 
~<  om  ->  x  ~<_  om )
1412, 13sylbi 187 . . . . . . 7  |-  ( x  e.  Fin  ->  x  ~<_  om )
1514ralimi 2631 . . . . . 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 8181 . . . . 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 8176 . . . . . 6  |-  ( T  e.  V  ->  ( F  Fn  T  ->  ran 
F  ~<_  T ) )
204, 3, 19sylc 56 . . . . 5  |-  ( ph  ->  ran  F  ~<_  T )
21 omex 7360 . . . . . 6  |-  om  e.  _V
2221xpdom1 6977 . . . . 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 6930 . . . 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 8204 . . . . . 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 6974 . . . 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 6930 . . 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 8200 . . 3  |-  ( om  ~<_  T  ->  ( T  X.  T )  ~~  T
)
3529, 34syl 15 . 2  |-  ( ph  ->  ( T  X.  T
)  ~~  T )
36 domentr 6936 . 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 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165   U.cuni 3843   class class class wbr 4039   omcom 4672    X. cxp 4703   ran crn 4706    Fn wfn 5266   -->wf 5267    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878   Fincfn 6879
This theorem is referenced by:  acsdomd  14300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-ac2 8105
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588  df-acn 7591  df-ac 7759
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