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Theorem eliman0 5877
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 5869 . . . . 5  |-  ( A F ( F `  A )  \/  ( F `  A )  =  (/) )
2 orcom 385 . . . . 5  |-  ( ( A F ( F `
 A )  \/  ( F `  A
)  =  (/) )  <->  ( ( F `  A )  =  (/)  \/  A F ( F `  A
) ) )
31, 2mpbi 208 . . . 4  |-  ( ( F `  A )  =  (/)  \/  A F ( F `  A ) )
43ori 373 . . 3  |-  ( -.  ( F `  A
)  =  (/)  ->  A F ( F `  A ) )
5 breq1 4442 . . . 4  |-  ( x  =  A  ->  (
x F ( F `
 A )  <->  A F
( F `  A
) ) )
65rspcev 3207 . . 3  |-  ( ( A  e.  B  /\  A F ( F `  A ) )  ->  E. x  e.  B  x F ( F `  A ) )
74, 6sylan2 472 . 2  |-  ( ( A  e.  B  /\  -.  ( F `  A
)  =  (/) )  ->  E. x  e.  B  x F ( F `  A ) )
8 fvex 5858 . . 3  |-  ( F `
 A )  e. 
_V
98elima 5330 . 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 366    /\ wa 367    = wceq 1398    e. wcel 1823   E.wrex 2805   (/)c0 3783   class class class wbr 4439   "cima 4991   ` cfv 5570
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-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
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-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-xp 4994  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fv 5578
This theorem is referenced by:  ovima0  6427
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