Theorem oprvalelrn 4030

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 4027	. . 3 |- (F Fn (A X. B) -> ran F = {z | E.x e. A E.y e. B z = (xFy)})
2	1	eleq2d 1538	. 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 3974	. . . . . . . 8 |- (xFy) e. V
4		eleq1 1531	. . . . . . . 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 1740	. . . . 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 1740	. . 3 |- (E.x e. A E.y e. B (xFy) = C -> C e. V)
10		eqeq1 1478	. . . . 5 |- (z = C -> (z = (xFy) <-> C = (xFy)))
11		eqcom 1474	. . . . 5 |- (C = (xFy) <-> (xFy) = C)
12	10, 11	syl6bb 535	. . . 4 |- (z = C -> (z = (xFy) <-> (xFy) = C))
13	12	2rexbidv 1678	. . 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 1899	. 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 535	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 954   e. wcel 956  {cab 1461  E.wrex 1643  Vcvv 1807   X. cxp 3163  ran crn 3166   Fn wfn 3172  (class class class)co 3954

This theorem is referenced by:  retopbas 7605  blssioo 7865  tgioo 7867  hhssnv 9073

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-opr 3956

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