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Theorem wdomd 7999
Description: Deduce weak dominance from an implicit onto function. (Contributed by Stefan O'Rear, 13-Feb-2015.)
Hypotheses
Ref Expression
wdomd.b  |-  ( ph  ->  B  e.  W )
wdomd.o  |-  ( (
ph  /\  x  e.  A )  ->  E. y  e.  B  x  =  X )
Assertion
Ref Expression
wdomd  |-  ( ph  ->  A  ~<_*  B )
Distinct variable groups:    x, A, y    x, B, y    ph, x, y    x, X
Allowed substitution hints:    W( x, y)    X( y)

Proof of Theorem wdomd
StepHypRef Expression
1 wdomd.b . . . 4  |-  ( ph  ->  B  e.  W )
2 abrexexg 6748 . . . 4  |-  ( B  e.  W  ->  { x  |  E. y  e.  B  x  =  X }  e.  _V )
31, 2syl 16 . . 3  |-  ( ph  ->  { x  |  E. y  e.  B  x  =  X }  e.  _V )
4 wdomd.o . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  E. y  e.  B  x  =  X )
54ex 432 . . . . 5  |-  ( ph  ->  ( x  e.  A  ->  E. y  e.  B  x  =  X )
)
65alrimiv 1724 . . . 4  |-  ( ph  ->  A. x ( x  e.  A  ->  E. y  e.  B  x  =  X ) )
7 ssab 3556 . . . 4  |-  ( A 
C_  { x  |  E. y  e.  B  x  =  X }  <->  A. x ( x  e.  A  ->  E. y  e.  B  x  =  X ) )
86, 7sylibr 212 . . 3  |-  ( ph  ->  A  C_  { x  |  E. y  e.  B  x  =  X }
)
93, 8ssexd 4584 . 2  |-  ( ph  ->  A  e.  _V )
109, 1, 4wdom2d 7998 1  |-  ( ph  ->  A  ~<_*  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wal 1396    = wceq 1398    e. wcel 1823   {cab 2439   E.wrex 2805   _Vcvv 3106    C_ wss 3461   class class class wbr 4439    ~<_* cwdom 7975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-wdom 7977
This theorem is referenced by:  hsmexlem2  8798  unxpwdom3  31280
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