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Theorem uniimadom 8161
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
Dummy variable  y is distinct from all other variables.

Proof of Theorem uniimadom
StepHypRef Expression
1 uniimadom.1 . . . . 5  |-  A  e. 
_V
21funimaex 5295 . . . 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 5535 . . . . . . . 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 4027 . . . . . . . . . 10  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  ~<_  B  <->  y  ~<_  B ) )
76biimpd 200 . . . . . . . . 9  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  ~<_  B  ->  y  ~<_  B ) )
87reximi 2651 . . . . . . . 8  |-  ( E. x  e.  A  ( F `  x )  =  y  ->  E. x  e.  A  ( ( F `  x )  ~<_  B  ->  y  ~<_  B ) )
9 r19.36av 2689 . . . . . . . 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 2626 . . 3  |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  A. y  e.  ( F " A
) y  ~<_  B )
15 unidom 8160 . . 3  |-  ( ( ( F " A
)  e.  _V  /\  A. y  e.  ( F
" A ) y  ~<_  B )  ->  U. ( F " A )  ~<_  ( ( F " A
)  X.  B ) )
163, 14, 15syl2anc 644 . 2  |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  U. ( F " A )  ~<_  ( ( F " A
)  X.  B ) )
17 imadomg 8154 . . . . 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 6956 . . . 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 6909 . 2  |-  ( ( U. ( F " A )  ~<_  ( ( F " A )  X.  B )  /\  ( ( F " A )  X.  B
)  ~<_  ( A  X.  B ) )  ->  U. ( F " A
)  ~<_  ( A  X.  B ) )
2416, 22, 23syl2anc 644 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 1624    e. wcel 1685   A.wral 2544   E.wrex 2545   _Vcvv 2789   U.cuni 3828   class class class wbr 4024    X. cxp 4686   "cima 4691   Fun wfun 5215   ` cfv 5221    ~<_ cdom 6856
This theorem is referenced by:  uniimadomf  8162
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-ac2 8084
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-er 6655  df-map 6769  df-en 6859  df-dom 6860  df-card 7567  df-acn 7570  df-ac 7738
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