Theorem eldmrexrnb 6366

Description: For any element in the domain of a function, there is an element in the range of the function which is the value of the function at that element. Because of the definition df-fv 5896 of the value of a function, the theorem is only valid in general if the empty set is not contained in the range of the function (the implication "to the right" is always valid). Indeed, with the definition df-fv 5896 of the value of a function, ( F ` Y ) = (/) may mean that the value of F at Y is the empty set or that F is not defined at Y. (Contributed by Alexander van der Vekens, 17-Dec-2017.)

Assertion

Ref	Expression
eldmrexrnb	|- ( ( Fun F /\ (/) e/ ran F ) -> ( Y e. dom F <-> E. x e. ran F x = ( F ` Y ) ) )

Distinct variable groups:   x, F   x, Y

Proof of Theorem eldmrexrnb

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldmrexrn 6365	. . 3 |- ( Fun F -> ( Y e. dom F -> E. x e. ran F x = ( F ` Y ) ) )
2	1	adantr 481	. 2 |- ( ( Fun F /\ (/) e/ ran F ) -> ( Y e. dom F -> E. x e. ran F x = ( F ` Y ) ) )
3		eleq1 2689	. . . . 5 |- ( x = ( F ` Y ) -> ( x e. ran F <-> ( F ` Y ) e. ran F ) )
4		elnelne2 2908	. . . . . . . . 9 |- ( ( ( F ` Y ) e. ran F /\ (/) e/ ran F ) -> ( F ` Y ) =/= (/) )
5		n0 3931	. . . . . . . . . 10 |- ( ( F ` Y ) =/= (/) <-> E. y y e. ( F ` Y ) )
6		elfvdm 6220	. . . . . . . . . . 11 |- ( y e. ( F ` Y ) -> Y e. dom F )
7	6	exlimiv 1858	. . . . . . . . . 10 |- ( E. y y e. ( F ` Y ) -> Y e. dom F )
8	5, 7	sylbi 207	. . . . . . . . 9 |- ( ( F ` Y ) =/= (/) -> Y e. dom F )
9	4, 8	syl 17	. . . . . . . 8 |- ( ( ( F ` Y ) e. ran F /\ (/) e/ ran F ) -> Y e. dom F )
10	9	expcom 451	. . . . . . 7 |- ( (/) e/ ran F -> ( ( F ` Y ) e. ran F -> Y e. dom F ) )
11	10	adantl 482	. . . . . 6 |- ( ( Fun F /\ (/) e/ ran F ) -> ( ( F ` Y ) e. ran F -> Y e. dom F ) )
12	11	com12 32	. . . . 5 |- ( ( F ` Y ) e. ran F -> ( ( Fun F /\ (/) e/ ran F ) -> Y e. dom F ) )
13	3, 12	syl6bi 243	. . . 4 |- ( x = ( F ` Y ) -> ( x e. ran F -> ( ( Fun F /\ (/) e/ ran F ) -> Y e. dom F ) ) )
14	13	com13 88	. . 3 |- ( ( Fun F /\ (/) e/ ran F ) -> ( x e. ran F -> ( x = ( F ` Y ) -> Y e. dom F ) ) )
15	14	rexlimdv 3030	. 2 |- ( ( Fun F /\ (/) e/ ran F ) -> ( E. x e. ran F x = ( F ` Y ) -> Y e. dom F ) )
16	2, 15	impbid 202	1 |- ( ( Fun F /\ (/) e/ ran F ) -> ( Y e. dom F <-> E. x e. ran F x = ( F ` Y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   e/ wnel 2897   E.wrex 2913   (/)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-nel 2898  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: (None)