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Theorem uniimadom 8050
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 5187 . . . 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 5426 . . . . . . . 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 3923 . . . . . . . . . 10  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  ~<_  B  <->  y  ~<_  B ) )
76biimpd 200 . . . . . . . . 9  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  ~<_  B  ->  y  ~<_  B ) )
87reximi 2612 . . . . . . . 8  |-  ( E. x  e.  A  ( F `  x )  =  y  ->  E. x  e.  A  ( ( F `  x )  ~<_  B  ->  y  ~<_  B ) )
9 r19.36av 2650 . . . . . . . 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 2587 . . 3  |-  ( ( Fun  F  /\  A. x  e.  A  ( F `  x )  ~<_  B )  ->  A. y  e.  ( F " A
) y  ~<_  B )
15 unidom 8049 . . 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 8043 . . . . 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 6846 . . . 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 6799 . 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 2509   E.wrex 2510   _Vcvv 2727   U.cuni 3727   class class class wbr 3920    X. cxp 4578   "cima 4583   Fun wfun 4586   ` cfv 4592    ~<_ cdom 6747
This theorem is referenced by:  uniimadomf  8051
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-ac2 7973
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-card 7456  df-acn 7459  df-ac 7627
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