Theorem fvelrnb 5681

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 2514	. . . 4 |- ( E. x e. A ( F ` x ) = B <-> E. x ( x e. A /\ ( F ` x ) = B ) )
2		19.41v 1949	. . . . 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 109	. . . . . . . . . 10 |- ( ( x e. A /\ ( F ` x ) = B ) -> x e. A )
4	3	anim1i 340	. . . . . . . . 9 |- ( ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> ( x e. A /\ F Fn A ) )
5	4	ancomd 267	. . . . . . . 8 |- ( ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> ( F Fn A /\ x e. A ) )
6		funfvex 5644	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
7	6	funfni 5423	. . . . . . . 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 110	. . . . . . . . 9 |- ( ( x e. A /\ ( F ` x ) = B ) -> ( F ` x ) = B )
10	9	eleq1d 2298	. . . . . . . 8 |- ( ( x e. A /\ ( F ` x ) = B ) -> ( ( F ` x ) e. _V <-> B e. _V ) )
11	10	adantr 276	. . . . . . 7 |- ( ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> ( ( F ` x ) e. _V <-> B e. _V ) )
12	8, 11	mpbid 147	. . . . . 6 |- ( ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> B e. _V )
13	12	exlimiv 1644	. . . . 5 |- ( E. x ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> B e. _V )
14	2, 13	sylbir 135	. . . 4 |- ( ( E. x ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> B e. _V )
15	1, 14	sylanb 284	. . 3 |- ( ( E. x e. A ( F ` x ) = B /\ F Fn A ) -> B e. _V )
16	15	expcom 116	. 2 |- ( F Fn A -> ( E. x e. A ( F ` x ) = B -> B e. _V ) )
17		fnrnfv 5680	. . . 4 |- ( F Fn A -> ran F = { y | E. x e. A y = ( F ` x ) } )
18	17	eleq2d 2299	. . 3 |- ( F Fn A -> ( B e. ran F <-> B e. { y | E. x e. A y = ( F ` x ) } ) )
19		eqeq1 2236	. . . . . 6 |- ( y = B -> ( y = ( F ` x ) <-> B = ( F ` x ) ) )
20		eqcom 2231	. . . . . 6 |- ( B = ( F ` x ) <-> ( F ` x ) = B )
21	19, 20	bitrdi 196	. . . . 5 |- ( y = B -> ( y = ( F ` x ) <-> ( F ` x ) = B ) )
22	21	rexbidv 2531	. . . 4 |- ( y = B -> ( E. x e. A y = ( F ` x ) <-> E. x e. A ( F ` x ) = B ) )
23	22	elab3g 2954	. . 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 463	. 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 422	1 |- ( F Fn A -> ( B e. ran F <-> E. x e. A ( F ` x ) = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   {cab 2215   E.wrex 2509   _Vcvv 2799   ran crn 4720   Fn wfn 5313   ` cfv 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326

This theorem is referenced by:  foelcdmi  5686  chfnrn  5746  rexrn  5772  ralrn  5773  elrnrexdmb  5775  ffnfv  5793  fconstfvm  5857  elunirn  5890  isoini  5942  canth  5952  reldm  6332  ordiso2  7202  eldju  7235  ctssdc  7280  uzn0  9738  frec2uzrand  10627  frecuzrdgtcl  10634  frecuzrdgfunlem  10641  uzin2  11498  imasmnd2  13485  imasgrp2  13647  imasrng  13919  imasring  14027  reeff1o  15447  uhgr2edg  16004  ushgredgedg  16024  ushgredgedgloop  16026