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Theorem eliman0 5895
Description: A non-nul function value is an element of the image of the function. (Contributed by Thierry Arnoux, 25-Jun-2019.)
Assertion
Ref Expression
eliman0  |-  ( ( A  e.  B  /\  -.  ( F `  A
)  =  (/) )  -> 
( F `  A
)  e.  ( F
" B ) )

Proof of Theorem eliman0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fvbr0 5887 . . . . 5  |-  ( A F ( F `  A )  \/  ( F `  A )  =  (/) )
2 orcom 387 . . . . 5  |-  ( ( A F ( F `
 A )  \/  ( F `  A
)  =  (/) )  <->  ( ( F `  A )  =  (/)  \/  A F ( F `  A
) ) )
31, 2mpbi 208 . . . 4  |-  ( ( F `  A )  =  (/)  \/  A F ( F `  A ) )
43ori 375 . . 3  |-  ( -.  ( F `  A
)  =  (/)  ->  A F ( F `  A ) )
5 breq1 4450 . . . 4  |-  ( x  =  A  ->  (
x F ( F `
 A )  <->  A F
( F `  A
) ) )
65rspcev 3214 . . 3  |-  ( ( A  e.  B  /\  A F ( F `  A ) )  ->  E. x  e.  B  x F ( F `  A ) )
74, 6sylan2 474 . 2  |-  ( ( A  e.  B  /\  -.  ( F `  A
)  =  (/) )  ->  E. x  e.  B  x F ( F `  A ) )
8 fvex 5876 . . 3  |-  ( F `
 A )  e. 
_V
98elima 5342 . 2  |-  ( ( F `  A )  e.  ( F " B )  <->  E. x  e.  B  x F
( F `  A
) )
107, 9sylibr 212 1  |-  ( ( A  e.  B  /\  -.  ( F `  A
)  =  (/) )  -> 
( F `  A
)  e.  ( F
" B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815   (/)c0 3785   class class class wbr 4447   "cima 5002   ` cfv 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fv 5596
This theorem is referenced by:  ovima0  6438
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