Theorem afvelrnb0 41244

Description: A member of a function's range is a value of the function, only one direction of implication of fvelrnb 6243. (Contributed by Alexander van der Vekens, 1-Jun-2017.)

Assertion

Ref	Expression
afvelrnb0	|- ( F Fn A -> ( B e. ran F -> E. x e. A ( F''' x ) = B ) )

Distinct variable groups:   x, A   x, B   x, F

Proof of Theorem afvelrnb0

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnrnafv 41242	. . 3 |- ( F Fn A -> ran F = { y | E. x e. A y = ( F''' x ) } )
2	1	eleq2d 2687	. 2 |- ( F Fn A -> ( B e. ran F <-> B e. { y | E. x e. A y = ( F''' x ) } ) )
3		eqeq1 2626	. . . . . 6 |- ( y = B -> ( y = ( F''' x ) <-> B = ( F''' x ) ) )
4		eqcom 2629	. . . . . 6 |- ( B = ( F''' x ) <-> ( F''' x ) = B )
5	3, 4	syl6bb 276	. . . . 5 |- ( y = B -> ( y = ( F''' x ) <-> ( F''' x ) = B ) )
6	5	rexbidv 3052	. . . 4 |- ( y = B -> ( E. x e. A y = ( F''' x ) <-> E. x e. A ( F''' x ) = B ) )
7	6	elabg 3351	. . 3 |- ( B e. { y | E. x e. A y = ( F''' x ) } -> ( B e. { y | E. x e. A y = ( F''' x ) } <-> E. x e. A ( F''' x ) = B ) )
8	7	ibi 256	. 2 |- ( B e. { y | E. x e. A y = ( F''' x ) } -> E. x e. A ( F''' x ) = B )
9	2, 8	syl6bi 243	1 |- ( F Fn A -> ( B e. ran F -> E. x e. A ( F''' x ) = B ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   ran crn 5115   Fn wfn 5883  '''cafv 41194

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  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-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-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-dfat 41196  df-afv 41197

This theorem is referenced by:  ffnafv  41251