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

Theorem uniimadom 8182
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 5346 . . . 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 5590 . . . . . . . 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 4042 . . . . . . . . . 10  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  ~<_  B  <->  y  ~<_  B ) )
76biimpd 198 . . . . . . . . 9  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  ~<_  B  ->  y  ~<_  B ) )
87reximi 2663 . . . . . . . 8  |-  ( E. x  e.  A  ( F `  x )  =  y  ->  E. x  e.  A  ( ( F `  x )  ~<_  B  ->  y  ~<_  B ) )
9 r19.36av 2701 . . . . . . . 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 2638 . . 3  |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  A. y  e.  ( F " A
) y  ~<_  B )
15 unidom 8181 . . 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 8175 . . . . 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 6977 . . . 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 6930 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801   U.cuni 3843   class class class wbr 4039    X. cxp 4703   "cima 4708   Fun wfun 5265   ` cfv 5271    ~<_ cdom 6877
This theorem is referenced by:  uniimadomf  8183
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-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-suc 4414  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-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-card 7588  df-acn 7591  df-ac 7759
  Copyright terms: Public domain W3C validator