Theorem elrnrexdm 5306

Description: For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 8-Dec-2017.)

Assertion

Ref	Expression
elrnrexdm	|- ( Fun F -> ( Y e. ran F -> E. x e. dom F Y = ( F ` x ) ) )

Distinct variable groups:   x, F   x, Y

Proof of Theorem elrnrexdm

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2041	. . . . . 6 |- ( Y e. ran F -> Y = Y )
2	1	ancli 306	. . . . 5 |- ( Y e. ran F -> ( Y e. ran F /\ Y = Y ) )
3	2	adantl 262	. . . 4 |- ( ( Fun F /\ Y e. ran F ) -> ( Y e. ran F /\ Y = Y ) )
4		eqeq2 2049	. . . . 5 |- ( y = Y -> ( Y = y <-> Y = Y ) )
5	4	rspcev 2656	. . . 4 |- ( ( Y e. ran F /\ Y = Y ) -> E. y e. ran F Y = y )
6	3, 5	syl 14	. . 3 |- ( ( Fun F /\ Y e. ran F ) -> E. y e. ran F Y = y )
7	6	ex 108	. 2 |- ( Fun F -> ( Y e. ran F -> E. y e. ran F Y = y ) )
8		funfn 4931	. . 3 |- ( Fun F <-> F Fn dom F )
9		eqeq2 2049	. . . 4 |- ( y = ( F ` x ) -> ( Y = y <-> Y = ( F ` x ) ) )
10	9	rexrn 5304	. . 3 |- ( F Fn dom F -> ( E. y e. ran F Y = y <-> E. x e. dom F Y = ( F ` x ) ) )
11	8, 10	sylbi 114	. 2 |- ( Fun F -> ( E. y e. ran F Y = y <-> E. x e. dom F Y = ( F ` x ) ) )
12	7, 11	sylibd 138	1 |- ( Fun F -> ( Y e. ran F -> E. x e. dom F Y = ( F ` x ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   E.wrex 2307   dom cdm 4345   ran crn 4346   Fun wfun 4896   Fn wfn 4897   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910

This theorem is referenced by: (None)