Theorem fvelrnb 5221

Description: A member of a function's range is a value of the function. (Contributed by NM, 31-Oct-1995.)

Assertion

Ref	Expression
fvelrnb	|- ( 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 fvelrnb

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rex 2312	. . . 4 |- ( E. x e. A ( F ` x ) = B <-> E. x ( x e. A /\ ( F ` x ) = B ) )
2		19.41v 1782	. . . . 5 |- ( E. x ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) <-> ( E. x ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) )
3		simpl 102	. . . . . . . . . 10 |- ( ( x e. A /\ ( F ` x ) = B ) -> x e. A )
4	3	anim1i 323	. . . . . . . . 9 |- ( ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> ( x e. A /\ F Fn A ) )
5	4	ancomd 254	. . . . . . . 8 |- ( ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> ( F Fn A /\ x e. A ) )
6		funfvex 5192	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
7	6	funfni 4999	. . . . . . . 8 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
8	5, 7	syl 14	. . . . . . 7 |- ( ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> ( F ` x ) e. _V )
9		simpr 103	. . . . . . . . 9 |- ( ( x e. A /\ ( F ` x ) = B ) -> ( F ` x ) = B )
10	9	eleq1d 2106	. . . . . . . 8 |- ( ( x e. A /\ ( F ` x ) = B ) -> ( ( F ` x ) e. _V <-> B e. _V ) )
11	10	adantr 261	. . . . . . 7 |- ( ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> ( ( F ` x ) e. _V <-> B e. _V ) )
12	8, 11	mpbid 135	. . . . . 6 |- ( ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> B e. _V )
13	12	exlimiv 1489	. . . . 5 |- ( E. x ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> B e. _V )
14	2, 13	sylbir 125	. . . 4 |- ( ( E. x ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> B e. _V )
15	1, 14	sylanb 268	. . 3 |- ( ( E. x e. A ( F ` x ) = B /\ F Fn A ) -> B e. _V )
16	15	expcom 109	. 2 |- ( F Fn A -> ( E. x e. A ( F ` x ) = B -> B e. _V ) )
17		fnrnfv 5220	. . . 4 |- ( F Fn A -> ran F = { y | E. x e. A y = ( F ` x ) } )
18	17	eleq2d 2107	. . 3 |- ( F Fn A -> ( B e. ran F <-> B e. { y | E. x e. A y = ( F ` x ) } ) )
19		eqeq1 2046	. . . . . 6 |- ( y = B -> ( y = ( F ` x ) <-> B = ( F ` x ) ) )
20		eqcom 2042	. . . . . 6 |- ( B = ( F ` x ) <-> ( F ` x ) = B )
21	19, 20	syl6bb 185	. . . . 5 |- ( y = B -> ( y = ( F ` x ) <-> ( F ` x ) = B ) )
22	21	rexbidv 2327	. . . 4 |- ( y = B -> ( E. x e. A y = ( F ` x ) <-> E. x e. A ( F ` x ) = B ) )
23	22	elab3g 2693	. . 3 |- ( ( E. x e. A ( F ` x ) = B -> B e. _V ) -> ( B e. { y | E. x e. A y = ( F ` x ) } <-> E. x e. A ( F ` x ) = B ) )
24	18, 23	sylan9bbr 436	. 2 |- ( ( ( E. x e. A ( F ` x ) = B -> B e. _V ) /\ F Fn A ) -> ( B e. ran F <-> E. x e. A ( F ` x ) = B ) )
25	16, 24	mpancom 399	1 |- ( F Fn A -> ( B e. ran F <-> E. x e. A ( F ` x ) = B ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   {cab 2026   E.wrex 2307   _Vcvv 2557   ran crn 4346   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:  chfnrn  5278  rexrn  5304  ralrn  5305  elrnrexdmb  5307  ffnfv  5323  fconstfvm  5379  elunirn  5405  isoini  5457  reldm  5812  ordiso2  6357  uzn0  8488  frec2uzrand  9191  frecuzrdgfn  9198  uzin2  9586