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

Theorem uniimadom 8120
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
StepHypRef Expression
1 uniimadom.1 . . . . 5  |-  A  e. 
_V
21funimaex 5254 . . . 4  |-  ( Fun 
F  ->  ( F " A )  e.  _V )
32adantr 453 . . 3  |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  ( F " A )  e. 
_V )
4 fvelima 5494 . . . . . . . 8  |-  ( ( Fun  F  /\  y  e.  ( F " A
) )  ->  E. x  e.  A  ( F `  x )  =  y )
54ex 425 . . . . . . 7  |-  ( Fun 
F  ->  ( y  e.  ( F " A
)  ->  E. x  e.  A  ( F `  x )  =  y ) )
6 breq1 3986 . . . . . . . . . 10  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  ~<_  B  <->  y  ~<_  B ) )
76biimpd 200 . . . . . . . . 9  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  ~<_  B  ->  y  ~<_  B ) )
87reximi 2623 . . . . . . . 8  |-  ( E. x  e.  A  ( F `  x )  =  y  ->  E. x  e.  A  ( ( F `  x )  ~<_  B  ->  y  ~<_  B ) )
9 r19.36av 2661 . . . . . . . 8  |-  ( E. x  e.  A  ( ( F `  x
)  ~<_  B  ->  y  ~<_  B )  ->  ( A. x  e.  A  ( F `  x )  ~<_  B  ->  y  ~<_  B ) )
108, 9syl 17 . . . . . . 7  |-  ( E. x  e.  A  ( F `  x )  =  y  ->  ( A. x  e.  A  ( F `  x )  ~<_  B  ->  y  ~<_  B ) )
115, 10syl6 31 . . . . . 6  |-  ( Fun 
F  ->  ( y  e.  ( F " A
)  ->  ( A. x  e.  A  ( F `  x )  ~<_  B  ->  y  ~<_  B ) ) )
1211com23 74 . . . . 5  |-  ( Fun 
F  ->  ( A. x  e.  A  ( F `  x )  ~<_  B  ->  ( y  e.  ( F " A
)  ->  y  ~<_  B ) ) )
1312imp 420 . . . 4  |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  (
y  e.  ( F
" A )  -> 
y  ~<_  B ) )
1413ralrimiv 2598 . . 3  |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  A. y  e.  ( F " A
) y  ~<_  B )
15 unidom 8119 . . 3  |-  ( ( ( F " A
)  e.  _V  /\  A. y  e.  ( F
" A ) y  ~<_  B )  ->  U. ( F " A )  ~<_  ( ( F " A
)  X.  B ) )
163, 14, 15syl2anc 645 . 2  |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  U. ( F " A )  ~<_  ( ( F " A
)  X.  B ) )
17 imadomg 8113 . . . . 5  |-  ( A  e.  _V  ->  ( Fun  F  ->  ( F " A )  ~<_  A ) )
181, 17ax-mp 10 . . . 4  |-  ( Fun 
F  ->  ( F " A )  ~<_  A )
19 uniimadom.2 . . . . 5  |-  B  e. 
_V
2019xpdom1 6915 . . . 4  |-  ( ( F " A )  ~<_  A  ->  ( ( F " A )  X.  B )  ~<_  ( A  X.  B ) )
2118, 20syl 17 . . 3  |-  ( Fun 
F  ->  ( ( F " A )  X.  B )  ~<_  ( A  X.  B ) )
2221adantr 453 . 2  |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  (
( F " A
)  X.  B )  ~<_  ( A  X.  B
) )
23 domtr 6868 . 2  |-  ( ( U. ( F " A )  ~<_  ( ( F " A )  X.  B )  /\  ( ( F " A )  X.  B
)  ~<_  ( A  X.  B ) )  ->  U. ( F " A
)  ~<_  ( A  X.  B ) )
2416, 22, 23syl2anc 645 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 6    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2516   E.wrex 2517   _Vcvv 2757   U.cuni 3787   class class class wbr 3983    X. cxp 4645   "cima 4650   Fun wfun 4653   ` cfv 4659    ~<_ cdom 6815
This theorem is referenced by:  uniimadomf  8121
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-ac2 8043
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-suc 4356  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-er 6614  df-map 6728  df-en 6818  df-dom 6819  df-card 7526  df-acn 7529  df-ac 7697
  Copyright terms: Public domain W3C validator