Theorem oprvalelrn 4045

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

Assertion

Ref	Expression
oprvalelrn	|- (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 oprvalelrn

Step	Hyp	Ref	Expression
1		fnrnoprval 4042	. . 3 |- (F Fn (A X. B) -> ran F = {z | E.x e. A E.y e. B z = (xFy)})
2	1	eleq2d 1544	. 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 3989	. . . . . . . 8 |- (xFy) e. V
4		eleq1 1537	. . . . . . . 8 |- ((xFy) = C -> ((xFy) e. V <-> C e. V))
5	3, 4	mpbii 193	. . . . . . 7 |- ((xFy) = C -> C e. V)
6	5	a1i 8	. . . . . 6 |- (y e. B -> ((xFy) = C -> C e. V))
7	6	r19.23aiv 1746	. . . . 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 1746	. . 3 |- (E.x e. A E.y e. B (xFy) = C -> C e. V)
10		eqeq1 1484	. . . . 5 |- (z = C -> (z = (xFy) <-> C = (xFy)))
11		eqcom 1480	. . . . 5 |- (C = (xFy) <-> (xFy) = C)
12	10, 11	syl6bb 538	. . . 4 |- (z = C -> (z = (xFy) <-> (xFy) = C))
13	12	2rexbidv 1684	. . 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 1906	. 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 538	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 146   = wceq 958   e. wcel 960  {cab 1466  E.wrex 1649  Vcvv 1814   X. cxp 3174  ran crn 3177   Fn wfn 3183  (class class class)co 3969

This theorem is referenced by:  retopbas 7652  blssioo 7910  tgioo 7912  hhssnv 9129

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-opr 3971

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