Theorem fvelrnb 3760

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 3759	. . 3 |- (F Fn A -> ran F = {y | E.x e. A y = (F` x)})
2	1	eleq2d 1541	. 2 |- (F Fn A -> (B e. ran F <-> B e. {y | E.x e. A y = (F` x)}))
3		fvex 3732	. . . . . 6 |- (F` x) e. V
4		eleq1 1534	. . . . . 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 1743	. . 3 |- (E.x e. A (F` x) = B -> B e. V)
8		eqeq1 1481	. . . . 5 |- (y = B -> (y = (F` x) <-> B = (F` x)))
9		eqcom 1477	. . . . 5 |- (B = (F` x) <-> (F` x) = B)
10	8, 9	syl6bb 536	. . . 4 |- (y = B -> (y = (F` x) <-> (F` x) = B))
11	10	rexbidv 1664	. . 3 |- (y = B -> (E.x e. A y = (F` x) <-> E.x e. A (F` x) = B))
12	7, 11	elab3 1903	. 2 |- (B e. {y | E.x e. A y = (F` x)} <-> E.x e. A (F` x) = B)
13	2, 12	syl6bb 536	1 |- (F Fn A -> (B e. ran F <-> E.x e. A (F` x) = B))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  {cab 1463  E.wrex 1646  Vcvv 1811  ran crn 3171   Fn wfn 3177  ` cfv 3182

This theorem is referenced by:  elrnopabg 3800  chfnrn 3802  ffnfv 3828  fconstfv 3849  elunirnALT 3869  isoini 3900  canth 3907  elrnoprabg 4124  mapenlem2 4490  inf0 4606  inf3lem6 4618  noinfep 4640  aceq5 4740  zorn2lem4 4791  isinfcard 4887  om2uzran 6300  fsequb2 6524  seq1ublem 6911  climsup 7155  cvgcmpub 7185  reeff1o 7426  unbenlem 7504  ruclem33 7542  ruclem35 7544  ruclem37 7546  ghgrpilem2 8134  ubthlem6 8534  bra11 10041  cnvbravalt 10043  pjssdif1 10103  pjhmopidm 10110  ghomgrpilem2 10386

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198

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