Theorem eliman0 6223

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.

Step	Hyp	Ref	Expression
1		fvbr0 6215	. . . . 5 |- ( A F ( F ` A ) \/ ( F ` A ) = (/) )
2		orcom 402	. . . . 5 |- ( ( A F ( F ` A ) \/ ( F ` A ) = (/) ) <-> ( ( F ` A ) = (/) \/ A F ( F ` A ) ) )
3	1, 2	mpbi 220	. . . 4 |- ( ( F ` A ) = (/) \/ A F ( F ` A ) )
4	3	ori 390	. . 3 |- ( -. ( F ` A ) = (/) -> A F ( F ` A ) )
5		breq1 4656	. . . 4 |- ( x = A -> ( x F ( F ` A ) <-> A F ( F ` A ) ) )
6	5	rspcev 3309	. . 3 |- ( ( A e. B /\ A F ( F ` A ) ) -> E. x e. B x F ( F ` A ) )
7	4, 6	sylan2 491	. 2 |- ( ( A e. B /\ -. ( F ` A ) = (/) ) -> E. x e. B x F ( F ` A ) )
8		fvex 6201	. . 3 |- ( F ` A ) e. _V
9	8	elima 5471	. 2 |- ( ( F ` A ) e. ( F " B ) <-> E. x e. B x F ( F ` A ) )
10	7, 9	sylibr 224	1 |- ( ( A e. B /\ -. ( F ` A ) = (/) ) -> ( F ` A ) e. ( F " B ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   (/)c0 3915   class class class wbr 4653   "cima 5117   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896

This theorem is referenced by:  ovima0  6813  setrec2fun  42439