Theorem elrnrexdmb 5625

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 5218	. . 3 |- ( Fun F <-> F Fn dom F )
2		fvelrnb 5534	. . 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 2167	. . 3 |- ( Y = ( F ` x ) <-> ( F ` x ) = Y )
5	4	rexbii 2473	. 2 |- ( E. x e. dom F Y = ( F ` x ) <-> E. x e. dom F ( F ` x ) = Y )
6	3, 5	bitr4di 197	1 |- ( Fun F -> ( Y e. ran F <-> E. x e. dom F Y = ( F ` x ) ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   e. wcel 2136   E.wrex 2445   dom cdm 4604   ran crn 4605   Fun wfun 5182   Fn wfn 5183   ` cfv 5188

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196

This theorem is referenced by: (None)