Theorem funeldmb 31661

Description: If (/) is not part of the range of a function F, then A is in the domain of F iff ( F ` A ) =/= (/). (Contributed by Scott Fenton, 7-Dec-2021.)

Assertion

Ref	Expression
funeldmb	|- ( ( Fun F /\ -. (/) e. ran F ) -> ( A e. dom F <-> ( F ` A ) =/= (/) ) )

Proof of Theorem funeldmb

Step	Hyp	Ref	Expression
1		fvelrn 6352	. . . . . . . 8 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. ran F )
2	1	ex 450	. . . . . . 7 |- ( Fun F -> ( A e. dom F -> ( F ` A ) e. ran F ) )
3	2	adantr 481	. . . . . 6 |- ( ( Fun F /\ ( F ` A ) = (/) ) -> ( A e. dom F -> ( F ` A ) e. ran F ) )
4		eleq1 2689	. . . . . . 7 |- ( ( F ` A ) = (/) -> ( ( F ` A ) e. ran F <-> (/) e. ran F ) )
5	4	adantl 482	. . . . . 6 |- ( ( Fun F /\ ( F ` A ) = (/) ) -> ( ( F ` A ) e. ran F <-> (/) e. ran F ) )
6	3, 5	sylibd 229	. . . . 5 |- ( ( Fun F /\ ( F ` A ) = (/) ) -> ( A e. dom F -> (/) e. ran F ) )
7	6	con3d 148	. . . 4 |- ( ( Fun F /\ ( F ` A ) = (/) ) -> ( -. (/) e. ran F -> -. A e. dom F ) )
8	7	impancom 456	. . 3 |- ( ( Fun F /\ -. (/) e. ran F ) -> ( ( F ` A ) = (/) -> -. A e. dom F ) )
9		ndmfv 6218	. . 3 |- ( -. A e. dom F -> ( F ` A ) = (/) )
10	8, 9	impbid1 215	. 2 |- ( ( Fun F /\ -. (/) e. ran F ) -> ( ( F ` A ) = (/) <-> -. A e. dom F ) )
11	10	necon2abid 2836	1 |- ( ( Fun F /\ -. (/) e. ran F ) -> ( A e. dom F <-> ( F ` A ) =/= (/) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   dom cdm 5114   ran crn 5115   Fun wfun 5882   ` 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-8 1992  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-pow 4843  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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  nosepssdm  31836