Theorem oprvelrn 4969

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

Assertion

Ref	Expression
oprvelrn	|- (F Fn (A X. B) -> (C e. ran F <-> E.x e. A E.y e. B (xFy) = C))

Distinct variable groups:   x,y,A   x,B,y   x,C,y   x,F,y

Proof of Theorem oprvelrn

Step	Hyp	Ref	Expression
1		fnrnoprv 4966	. . 3 |- (F Fn (A X. B) -> ran F = {z | E.x e. A E.y e. B z = (xFy)})
2	1	eleq2d 1964	. 2 |- (F Fn (A X. B) -> (C e. ran F <-> C e. {z | E.x e. A E.y e. B z = (xFy)}))
3		oprex 4907	. . . . . . . 8 |- (xFy) e. _V
4		eleq1 1957	. . . . . . . 8 |- ((xFy) = C -> ((xFy) e. _V <-> C e. _V))
5	3, 4	mpbii 210	. . . . . . 7 |- ((xFy) = C -> C e. _V)
6	5	a1i 8	. . . . . 6 |- (y e. B -> ((xFy) = C -> C e. _V))
7	6	r19.23aiv 2211	. . . . 5 |- (E.y e. B (xFy) = C -> C e. _V)
8	7	a1i 8	. . . 4 |- (x e. A -> (E.y e. B (xFy) = C -> C e. _V))
9	8	r19.23aiv 2211	. . 3 |- (E.x e. A E.y e. B (xFy) = C -> C e. _V)
10		eqeq1 1890	. . . . 5 |- (z = C -> (z = (xFy) <-> C = (xFy)))
11		eqcom 1886	. . . . 5 |- (C = (xFy) <-> (xFy) = C)
12	10, 11	syl6bb 595	. . . 4 |- (z = C -> (z = (xFy) <-> (xFy) = C))
13	12	2rexbidv 2141	. . 3 |- (z = C -> (E.x e. A E.y e. B z = (xFy) <-> E.x e. A E.y e. B (xFy) = C))
14	9, 13	elab3 2412	. 2 |- (C e. {z | E.x e. A E.y e. B z = (xFy)} <-> E.x e. A E.y e. B (xFy) = C)
15	2, 14	syl6bb 595	1 |- (F Fn (A X. B) -> (C e. ran F <-> E.x e. A E.y e. B (xFy) = C))

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  {cab 1871  E.wrex 2106  _Vcvv 2292   X. cxp 3984  ran crn 3987   Fn wfn 3993  (class class class)co 4884

This theorem is referenced by:  retopbas 8925  blssioo 9191  tgioo 9193  hhssnv 10767  fictb 15371  heiborlem29 15983

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886

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