Theorem fvelrnb 5626

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 2490	. . . 4 |- ( E. x e. A ( F ` x ) = B <-> E. x ( x e. A /\ ( F ` x ) = B ) )
2		19.41v 1926	. . . . 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 5593	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
7	6	funfni 5376	. . . . . . . 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 2274	. . . . . . . 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 1621	. . . . 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 5625	. . . 4 |- ( F Fn A -> ran F = { y | E. x e. A y = ( F ` x ) } )
18	17	eleq2d 2275	. . 3 |- ( F Fn A -> ( B e. ran F <-> B e. { y | E. x e. A y = ( F ` x ) } ) )
19		eqeq1 2212	. . . . . 6 |- ( y = B -> ( y = ( F ` x ) <-> B = ( F ` x ) ) )
20		eqcom 2207	. . . . . 6 |- ( B = ( F ` x ) <-> ( F ` x ) = B )
21	19, 20	bitrdi 196	. . . . 5 |- ( y = B -> ( y = ( F ` x ) <-> ( F ` x ) = B ) )
22	21	rexbidv 2507	. . . 4 |- ( y = B -> ( E. x e. A y = ( F ` x ) <-> E. x e. A ( F ` x ) = B ) )
23	22	elab3g 2924	. . 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 1373   E.wex 1515   e. wcel 2176   {cab 2191   E.wrex 2485   _Vcvv 2772   ran crn 4676   Fn wfn 5266   ` cfv 5271

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279

This theorem is referenced by:  foelcdmi  5631  chfnrn  5691  rexrn  5717  ralrn  5718  elrnrexdmb  5720  ffnfv  5738  fconstfvm  5802  elunirn  5835  isoini  5887  canth  5897  reldm  6272  ordiso2  7137  eldju  7170  ctssdc  7215  uzn0  9664  frec2uzrand  10550  frecuzrdgtcl  10557  frecuzrdgfunlem  10564  uzin2  11298  imasmnd2  13284  imasgrp2  13446  imasrng  13718  imasring  13826  reeff1o  15245