Theorem elrnrexdmb 5560

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, 17-Dec-2017.)

Assertion

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

Distinct variable groups:   x, F   x, Y

Proof of Theorem elrnrexdmb

Step	Hyp	Ref	Expression
1		funfn 5153	. . 3 |- ( Fun F <-> F Fn dom F )
2		fvelrnb 5469	. . 3 |- ( F Fn dom F -> ( Y e. ran F <-> E. x e. dom F ( F ` x ) = Y ) )
3	1, 2	sylbi 120	. 2 |- ( Fun F -> ( Y e. ran F <-> E. x e. dom F ( F ` x ) = Y ) )
4		eqcom 2141	. . 3 |- ( Y = ( F ` x ) <-> ( F ` x ) = Y )
5	4	rexbii 2442	. 2 |- ( E. x e. dom F Y = ( F ` x ) <-> E. x e. dom F ( F ` x ) = Y )
6	3, 5	syl6bbr 197	1 |- ( Fun F -> ( Y e. ran F <-> E. x e. dom F Y = ( F ` x ) ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   e. wcel 1480   E.wrex 2417   dom cdm 4539   ran crn 4540   Fun wfun 5117   Fn wfn 5118   ` cfv 5123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131

This theorem is referenced by: (None)