Theorem elrnrexdmb 5795

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 5363	. . 3 |- ( Fun F <-> F Fn dom F )
2		fvelrnb 5702	. . 3 |- ( F Fn dom F -> ( Y e. ran F <-> E. x e. dom F ( F ` x ) = Y ) )
3	1, 2	sylbi 121	. 2 |- ( Fun F -> ( Y e. ran F <-> E. x e. dom F ( F ` x ) = Y ) )
4		eqcom 2233	. . 3 |- ( Y = ( F ` x ) <-> ( F ` x ) = Y )
5	4	rexbii 2540	. 2 |- ( E. x e. dom F Y = ( F ` x ) <-> E. x e. dom F ( F ` x ) = Y )
6	3, 5	bitr4di 198	1 |- ( Fun F -> ( Y e. ran F <-> E. x e. dom F Y = ( F ` x ) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2202   E.wrex 2512   dom cdm 4731   ran crn 4732   Fun wfun 5327   Fn wfn 5328   ` cfv 5333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341

This theorem is referenced by:  edgiedgbg  15989  uhgrspansubgrlem  16200