Theorem fvelrnb 3745

Description: A member of a function's range is a value of the function.

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

Step	Hyp	Ref	Expression
1		fnrnfv 3744	. . 3 |- (F Fn A -> ran F = {y | E.x e. A y = (F` x)})
2	1	eleq2d 1533	. 2 |- (F Fn A -> (B e. ran F <-> B e. {y | E.x e. A y = (F` x)}))
3		fvex 3717	. . . . . 6 |- (F` x) e. V
4		eleq1 1526	. . . . . 6 |- ((F` x) = B -> ((F` x) e. V <-> B e. V))
5	3, 4	mpbii 193	. . . . 5 |- ((F` x) = B -> B e. V)
6	5	a1i 8	. . . 4 |- (x e. A -> ((F` x) = B -> B e. V))
7	6	r19.23aiv 1735	. . 3 |- (E.x e. A (F` x) = B -> B e. V)
8		eqeq1 1473	. . . . 5 |- (y = B -> (y = (F` x) <-> B = (F` x)))
9		eqcom 1469	. . . . 5 |- (B = (F` x) <-> (F` x) = B)
10	8, 9	syl6bb 534	. . . 4 |- (y = B -> (y = (F` x) <-> (F` x) = B))
11	10	rexbidv 1656	. . 3 |- (y = B -> (E.x e. A y = (F` x) <-> E.x e. A (F` x) = B))
12	7, 11	elab3 1894	. 2 |- (B e. {y | E.x e. A y = (F` x)} <-> E.x e. A (F` x) = B)
13	2, 12	syl6bb 534	1 |- (F Fn A -> (B e. ran F <-> E.x e. A (F` x) = B))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 953   e. wcel 955  {cab 1456  E.wrex 1638  Vcvv 1802  ran crn 3161   Fn wfn 3167  ` cfv 3172

This theorem is referenced by:  elrnopabg 3785  chfnrn 3787  ffnfv 3813  fconstfv 3834  elunirnALT 3854  isoini 3885  canth 3892  elrnoprabg 4108  mapenlem2 4470  inf0 4578  inf3lem6 4590  noinfep 4612  aceq5 4712  zorn2lem4 4763  isinfcard 4859  om2uzran 6237  fsequb2 6456  seq1ublem 6848  climsup 7091  cvgcmpub 7121  reeff1o 7368  unbenlem 7447  ruclem33 7485  ruclem35 7487  ruclem37 7489  ghgrpilem2 8071  ubthlem6 8465  bra11 9954  cnvbravalt 9956  pjssdif1 10014  pjhmopidm 10020  ghomgrpilem2 10291

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188

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