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

Theorem uniimadom 8166
Description: An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. (Contributed by NM, 25-Mar-2006.)
Hypotheses
Ref Expression
uniimadom.1  |-  A  e. 
_V
uniimadom.2  |-  B  e. 
_V
Assertion
Ref Expression
uniimadom  |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  U. ( F " A )  ~<_  ( A  X.  B ) )
Distinct variable groups:    x, A    x, B    x, F

Proof of Theorem uniimadom
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 uniimadom.1 . . . . 5  |-  A  e. 
_V
21funimaex 5330 . . . 4  |-  ( Fun 
F  ->  ( F " A )  e.  _V )
32adantr 451 . . 3  |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  ( F " A )  e. 
_V )
4 fvelima 5574 . . . . . . . 8  |-  ( ( Fun  F  /\  y  e.  ( F " A
) )  ->  E. x  e.  A  ( F `  x )  =  y )
54ex 423 . . . . . . 7  |-  ( Fun 
F  ->  ( y  e.  ( F " A
)  ->  E. x  e.  A  ( F `  x )  =  y ) )
6 breq1 4026 . . . . . . . . . 10  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  ~<_  B  <->  y  ~<_  B ) )
76biimpd 198 . . . . . . . . 9  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  ~<_  B  ->  y  ~<_  B ) )
87reximi 2650 . . . . . . . 8  |-  ( E. x  e.  A  ( F `  x )  =  y  ->  E. x  e.  A  ( ( F `  x )  ~<_  B  ->  y  ~<_  B ) )
9 r19.36av 2688 . . . . . . . 8  |-  ( E. x  e.  A  ( ( F `  x
)  ~<_  B  ->  y  ~<_  B )  ->  ( A. x  e.  A  ( F `  x )  ~<_  B  ->  y  ~<_  B ) )
108, 9syl 15 . . . . . . 7  |-  ( E. x  e.  A  ( F `  x )  =  y  ->  ( A. x  e.  A  ( F `  x )  ~<_  B  ->  y  ~<_  B ) )
115, 10syl6 29 . . . . . 6  |-  ( Fun 
F  ->  ( y  e.  ( F " A
)  ->  ( A. x  e.  A  ( F `  x )  ~<_  B  ->  y  ~<_  B ) ) )
1211com23 72 . . . . 5  |-  ( Fun 
F  ->  ( A. x  e.  A  ( F `  x )  ~<_  B  ->  ( y  e.  ( F " A
)  ->  y  ~<_  B ) ) )
1312imp 418 . . . 4  |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  (
y  e.  ( F
" A )  -> 
y  ~<_  B ) )
1413ralrimiv 2625 . . 3  |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  A. y  e.  ( F " A
) y  ~<_  B )
15 unidom 8165 . . 3  |-  ( ( ( F " A
)  e.  _V  /\  A. y  e.  ( F
" A ) y  ~<_  B )  ->  U. ( F " A )  ~<_  ( ( F " A
)  X.  B ) )
163, 14, 15syl2anc 642 . 2  |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  U. ( F " A )  ~<_  ( ( F " A
)  X.  B ) )
17 imadomg 8159 . . . . 5  |-  ( A  e.  _V  ->  ( Fun  F  ->  ( F " A )  ~<_  A ) )
181, 17ax-mp 8 . . . 4  |-  ( Fun 
F  ->  ( F " A )  ~<_  A )
19 uniimadom.2 . . . . 5  |-  B  e. 
_V
2019xpdom1 6961 . . . 4  |-  ( ( F " A )  ~<_  A  ->  ( ( F " A )  X.  B )  ~<_  ( A  X.  B ) )
2118, 20syl 15 . . 3  |-  ( Fun 
F  ->  ( ( F " A )  X.  B )  ~<_  ( A  X.  B ) )
2221adantr 451 . 2  |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  (
( F " A
)  X.  B )  ~<_  ( A  X.  B
) )
23 domtr 6914 . 2  |-  ( ( U. ( F " A )  ~<_  ( ( F " A )  X.  B )  /\  ( ( F " A )  X.  B
)  ~<_  ( A  X.  B ) )  ->  U. ( F " A
)  ~<_  ( A  X.  B ) )
2416, 22, 23syl2anc 642 1  |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  U. ( F " A )  ~<_  ( A  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788   U.cuni 3827   class class class wbr 4023    X. cxp 4687   "cima 4692   Fun wfun 5249   ` cfv 5255    ~<_ cdom 6861
This theorem is referenced by:  uniimadomf  8167
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-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-suc 4398  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-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-card 7572  df-acn 7575  df-ac 7743
  Copyright terms: Public domain W3C validator