MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unirnfdomd Structured version   Unicode version

Theorem unirnfdomd 8933
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 5724 . . . . . . . 8  |-  ( F : T --> Fin  ->  F  Fn  T )
31, 2syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  T )
4 unirnfdomd.3 . . . . . . 7  |-  ( ph  ->  T  e.  V )
5 fnex 6120 . . . . . . 7  |-  ( ( F  Fn  T  /\  T  e.  V )  ->  F  e.  _V )
63, 4, 5syl2anc 661 . . . . . 6  |-  ( ph  ->  F  e.  _V )
7 rnexg 6708 . . . . . 6  |-  ( F  e.  _V  ->  ran  F  e.  _V )
86, 7syl 16 . . . . 5  |-  ( ph  ->  ran  F  e.  _V )
9 frn 5730 . . . . . . 7  |-  ( F : T --> Fin  ->  ran 
F  C_  Fin )
10 dfss3 3489 . . . . . . 7  |-  ( ran 
F  C_  Fin  <->  A. x  e.  ran  F  x  e. 
Fin )
119, 10sylib 196 . . . . . 6  |-  ( F : T --> Fin  ->  A. x  e.  ran  F  x  e.  Fin )
12 isfinite 8060 . . . . . . . 8  |-  ( x  e.  Fin  <->  x  ~<  om )
13 sdomdom 7535 . . . . . . . 8  |-  ( x 
~<  om  ->  x  ~<_  om )
1412, 13sylbi 195 . . . . . . 7  |-  ( x  e.  Fin  ->  x  ~<_  om )
1514ralimi 2852 . . . . . 6  |-  ( A. x  e.  ran  F  x  e.  Fin  ->  A. x  e.  ran  F  x  ~<_  om )
161, 11, 153syl 20 . . . . 5  |-  ( ph  ->  A. x  e.  ran  F  x  ~<_  om )
17 unidom 8909 . . . . 5  |-  ( ( ran  F  e.  _V  /\ 
A. x  e.  ran  F  x  ~<_  om )  ->  U. ran  F  ~<_  ( ran  F  X.  om ) )
188, 16, 17syl2anc 661 . . . 4  |-  ( ph  ->  U. ran  F  ~<_  ( ran  F  X.  om ) )
19 fnrndomg 8904 . . . . . 6  |-  ( T  e.  V  ->  ( F  Fn  T  ->  ran 
F  ~<_  T ) )
204, 3, 19sylc 60 . . . . 5  |-  ( ph  ->  ran  F  ~<_  T )
21 omex 8051 . . . . . 6  |-  om  e.  _V
2221xpdom1 7608 . . . . 5  |-  ( ran 
F  ~<_  T  ->  ( ran  F  X.  om )  ~<_  ( T  X.  om )
)
2320, 22syl 16 . . . 4  |-  ( ph  ->  ( ran  F  X.  om )  ~<_  ( T  X.  om ) )
24 domtr 7560 . . . 4  |-  ( ( U. ran  F  ~<_  ( ran  F  X.  om )  /\  ( ran  F  X.  om )  ~<_  ( T  X.  om ) )  ->  U. ran  F  ~<_  ( T  X.  om )
)
2518, 23, 24syl2anc 661 . . 3  |-  ( ph  ->  U. ran  F  ~<_  ( T  X.  om )
)
26 unirnfdomd.2 . . . . 5  |-  ( ph  ->  -.  T  e.  Fin )
27 infinf 8932 . . . . . 6  |-  ( T  e.  V  ->  ( -.  T  e.  Fin  <->  om  ~<_  T ) )
284, 27syl 16 . . . . 5  |-  ( ph  ->  ( -.  T  e. 
Fin 
<->  om  ~<_  T ) )
2926, 28mpbid 210 . . . 4  |-  ( ph  ->  om  ~<_  T )
30 xpdom2g 7605 . . . 4  |-  ( ( T  e.  V  /\  om  ~<_  T )  ->  ( T  X.  om )  ~<_  ( T  X.  T ) )
314, 29, 30syl2anc 661 . . 3  |-  ( ph  ->  ( T  X.  om )  ~<_  ( T  X.  T ) )
32 domtr 7560 . . 3  |-  ( ( U. ran  F  ~<_  ( T  X.  om )  /\  ( T  X.  om )  ~<_  ( T  X.  T ) )  ->  U. ran  F  ~<_  ( T  X.  T ) )
3325, 31, 32syl2anc 661 . 2  |-  ( ph  ->  U. ran  F  ~<_  ( T  X.  T ) )
34 infxpidm 8928 . . 3  |-  ( om  ~<_  T  ->  ( T  X.  T )  ~~  T
)
3529, 34syl 16 . 2  |-  ( ph  ->  ( T  X.  T
)  ~~  T )
36 domentr 7566 . 2  |-  ( ( U. ran  F  ~<_  ( T  X.  T )  /\  ( T  X.  T )  ~~  T
)  ->  U. ran  F  ~<_  T )
3733, 35, 36syl2anc 661 1  |-  ( ph  ->  U. ran  F  ~<_  T )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    e. wcel 1762   A.wral 2809   _Vcvv 3108    C_ wss 3471   U.cuni 4240   class class class wbr 4442    X. cxp 4992   ran crn 4995    Fn wfn 5576   -->wf 5577   omcom 6673    ~~ cen 7505    ~<_ cdom 7506    ~< csdm 7507   Fincfn 7508
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 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-ac2 8834
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 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-oi 7926  df-card 8311  df-acn 8314  df-ac 8488
This theorem is referenced by:  acsdomd  15659
  Copyright terms: Public domain W3C validator