Theorem fvelrnb 5242

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 2354	. . . 4 |- ( E. x e. A ( F ` x ) = B <-> E. x ( x e. A /\ ( F ` x ) = B ) )
2		19.41v 1823	. . . . 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 107	. . . . . . . . . 10 |- ( ( x e. A /\ ( F ` x ) = B ) -> x e. A )
4	3	anim1i 333	. . . . . . . . 9 |- ( ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> ( x e. A /\ F Fn A ) )
5	4	ancomd 263	. . . . . . . 8 |- ( ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> ( F Fn A /\ x e. A ) )
6		funfvex 5212	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
7	6	funfni 5019	. . . . . . . 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 108	. . . . . . . . 9 |- ( ( x e. A /\ ( F ` x ) = B ) -> ( F ` x ) = B )
10	9	eleq1d 2147	. . . . . . . 8 |- ( ( x e. A /\ ( F ` x ) = B ) -> ( ( F ` x ) e. _V <-> B e. _V ) )
11	10	adantr 270	. . . . . . 7 |- ( ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> ( ( F ` x ) e. _V <-> B e. _V ) )
12	8, 11	mpbid 145	. . . . . 6 |- ( ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> B e. _V )
13	12	exlimiv 1529	. . . . 5 |- ( E. x ( ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> B e. _V )
14	2, 13	sylbir 133	. . . 4 |- ( ( E. x ( x e. A /\ ( F ` x ) = B ) /\ F Fn A ) -> B e. _V )
15	1, 14	sylanb 278	. . 3 |- ( ( E. x e. A ( F ` x ) = B /\ F Fn A ) -> B e. _V )
16	15	expcom 114	. 2 |- ( F Fn A -> ( E. x e. A ( F ` x ) = B -> B e. _V ) )
17		fnrnfv 5241	. . . 4 |- ( F Fn A -> ran F = { y | E. x e. A y = ( F ` x ) } )
18	17	eleq2d 2148	. . 3 |- ( F Fn A -> ( B e. ran F <-> B e. { y | E. x e. A y = ( F ` x ) } ) )
19		eqeq1 2087	. . . . . 6 |- ( y = B -> ( y = ( F ` x ) <-> B = ( F ` x ) ) )
20		eqcom 2083	. . . . . 6 |- ( B = ( F ` x ) <-> ( F ` x ) = B )
21	19, 20	syl6bb 194	. . . . 5 |- ( y = B -> ( y = ( F ` x ) <-> ( F ` x ) = B ) )
22	21	rexbidv 2369	. . . 4 |- ( y = B -> ( E. x e. A y = ( F ` x ) <-> E. x e. A ( F ` x ) = B ) )
23	22	elab3g 2744	. . 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 450	. 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 413	1 |- ( F Fn A -> ( B e. ran F <-> E. x e. A ( F ` x ) = B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   {cab 2067   E.wrex 2349   _Vcvv 2601   ran crn 4364   Fn wfn 4917   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930

This theorem is referenced by:  chfnrn  5299  rexrn  5325  ralrn  5326  elrnrexdmb  5328  ffnfv  5344  fconstfvm  5400  elunirn  5426  isoini  5477  reldm  5832  ordiso2  6446  uzn0  8634  frec2uzrand  9407  frecuzrdgfn  9414  uzin2  9873