Theorem elrnoprabg 4124

Description: Membership in the range of an operation class abstraction.

Hypothesis

Ref	Expression
elrnoprabg.1	|- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}

Assertion

Ref	Expression
elrnoprabg	|- (A.x e. A A.y e. B C e. R -> (D e. ran F <-> E.x e. A E.y e. B D = C))

Distinct variable groups:   x,y,z,A   x,B,y,z   z,C   x,D,y

Proof of Theorem elrnoprabg

Step	Hyp	Ref	Expression
1		elisset 1817	. . . . 5 |- (C e. R -> C e. V)
2	1	r19.20si 1706	. . . 4 |- (A.y e. B C e. R -> A.y e. B C e. V)
3	2	r19.20si 1706	. . 3 |- (A.x e. A A.y e. B C e. R -> A.x e. A A.y e. B C e. V)
4		elrnoprabg.1	. . . 4 |- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}
5	4	fnoprab2g 4121	. . 3 |- (A.x e. A A.y e. B C e. V <-> F Fn (A X. B))
6	3, 5	sylib 198	. 2 |- (A.x e. A A.y e. B C e. R -> F Fn (A X. B))
7		fvelrnb 3760	. . 3 |- (F Fn (A X. B) -> (D e. ran F <-> E.w e. (A X. B)(F` w) = D))
8		hbra1 1687	. . . . 5 |- (A.x e. A A.y e. B C e. R -> A.xA.x e. A A.y e. B C e. R)
9		ax-17 971	. . . . . . . 8 |- (x e. A -> A.y x e. A)
10		hbra1 1687	. . . . . . . 8 |- (A.y e. B C e. R -> A.yA.y e. B C e. R)
11	9, 10	hbral 1686	. . . . . . 7 |- (A.x e. A A.y e. B C e. R -> A.yA.x e. A A.y e. B C e. R)
12	11, 9	hban 1009	. . . . . 6 |- ((A.x e. A A.y e. B C e. R /\ x e. A) -> A.y(A.x e. A A.y e. B C e. R /\ x e. A))
13		ra42 1696	. . . . . . . . 9 |- (A.x e. A A.y e. B C e. R -> ((x e. A /\ y e. B) -> C e. R))
14	4	oprabval4g 4031	. . . . . . . . . . . 12 |- ((x e. A /\ y e. B /\ C e. R) -> (xFy) = C)
15	14	eqeq1d 1483	. . . . . . . . . . 11 |- ((x e. A /\ y e. B /\ C e. R) -> ((xFy) = D <-> C = D))
16		eqcom 1477	. . . . . . . . . . 11 |- (C = D <-> D = C)
17	15, 16	syl6bb 536	. . . . . . . . . 10 |- ((x e. A /\ y e. B /\ C e. R) -> ((xFy) = D <-> D = C))
18	17	3expia 835	. . . . . . . . 9 |- ((x e. A /\ y e. B) -> (C e. R -> ((xFy) = D <-> D = C)))
19	13, 18	sylcom 51	. . . . . . . 8 |- (A.x e. A A.y e. B C e. R -> ((x e. A /\ y e. B) -> ((xFy) = D <-> D = C)))
20	19	exp3a 375	. . . . . . 7 |- (A.x e. A A.y e. B C e. R -> (x e. A -> (y e. B -> ((xFy) = D <-> D = C))))
21	20	imp31 362	. . . . . 6 |- (((A.x e. A A.y e. B C e. R /\ x e. A) /\ y e. B) -> ((xFy) = D <-> D = C))
22	12, 21	rexbida 1658	. . . . 5 |- ((A.x e. A A.y e. B C e. R /\ x e. A) -> (E.y e. B (xFy) = D <-> E.y e. B D = C))
23	8, 22	rexbida 1658	. . . 4 |- (A.x e. A A.y e. B C e. R -> (E.x e. A E.y e. B (xFy) = D <-> E.x e. A E.y e. B D = C))
24		fveq2 3724	. . . . . . . 8 |- (w = <.v, u>. -> (F` w) = (F` <.v, u>.))
25		df-opr 3965	. . . . . . . 8 |- (vFu) = (F` <.v, u>.)
26	24, 25	syl6eqr 1525	. . . . . . 7 |- (w = <.v, u>. -> (F` w) = (vFu))
27	26	eqeq1d 1483	. . . . . 6 |- (w = <.v, u>. -> ((F` w) = D <-> (vFu) = D))
28	27	rexxp 3219	. . . . 5 |- (E.w e. (A X. B)(F` w) = D <-> E.v e. A E.u e. B (vFu) = D)
29		ax-17 971	. . . . . . 7 |- (u e. B -> A.x u e. B)
30		ax-17 971	. . . . . . . . 9 |- (w e. v -> A.x w e. v)
31		hboprab1 3993	. . . . . . . . . 10 |- (w e. {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} -> A.x w e. {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)})
32	4, 31	hbxfr 1563	. . . . . . . . 9 |- (w e. F -> A.x w e. F)
33		ax-17 971	. . . . . . . . 9 |- (w e. u -> A.x w e. u)
34	30, 32, 33	hbopr 3981	. . . . . . . 8 |- (w e. (vFu) -> A.x w e. (vFu))
35		ax-17 971	. . . . . . . 8 |- (w e. D -> A.x w e. D)
36	34, 35	hbeq 1565	. . . . . . 7 |- ((vFu) = D -> A.x(vFu) = D)
37	29, 36	hbrex 1688	. . . . . 6 |- (E.u e. B (vFu) = D -> A.xE.u e. B (vFu) = D)
38		ax-17 971	. . . . . 6 |- (E.u e. B (xFu) = D -> A.vE.u e. B (xFu) = D)
39		opreq1 3968	. . . . . . . 8 |- (v = x -> (vFu) = (xFu))
40	39	eqeq1d 1483	. . . . . . 7 |- (v = x -> ((vFu) = D <-> (xFu) = D))
41	40	rexbidv 1664	. . . . . 6 |- (v = x -> (E.u e. B (vFu) = D <-> E.u e. B (xFu) = D))
42	37, 38, 41	cbvrex 1799	. . . . 5 |- (E.v e. A E.u e. B (vFu) = D <-> E.x e. A E.u e. B (xFu) = D)
43		ax-17 971	. . . . . . . . 9 |- (w e. x -> A.y w e. x)
44		hboprab2 3994	. . . . . . . . . 10 |- (w e. {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} -> A.y w e. {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)})
45	4, 44	hbxfr 1563	. . . . . . . . 9 |- (w e. F -> A.y w e. F)
46		ax-17 971	. . . . . . . . 9 |- (w e. u -> A.y w e. u)
47	43, 45, 46	hbopr 3981	. . . . . . . 8 |- (w e. (xFu) -> A.y w e. (xFu))
48		ax-17 971	. . . . . . . 8 |- (w e. D -> A.y w e. D)
49	47, 48	hbeq 1565	. . . . . . 7 |- ((xFu) = D -> A.y(xFu) = D)
50		ax-17 971	. . . . . . 7 |- ((xFy) = D -> A.u(xFy) = D)
51		opreq2 3969	. . . . . . . 8 |- (u = y -> (xFu) = (xFy))
52	51	eqeq1d 1483	. . . . . . 7 |- (u = y -> ((xFu) = D <-> (xFy) = D))
53	49, 50, 52	cbvrex 1799	. . . . . 6 |- (E.u e. B (xFu) = D <-> E.y e. B (xFy) = D)
54	53	rexbii 1668	. . . . 5 |- (E.x e. A E.u e. B (xFu) = D <-> E.x e. A E.y e. B (xFy) = D)
55	28, 42, 54	3bitr 177	. . . 4 |- (E.w e. (A X. B)(F` w) = D <-> E.x e. A E.y e. B (xFy) = D)
56	23, 55	syl5bb 532	. . 3 |- (A.x e. A A.y e. B C e. R -> (E.w e. (A X. B)(F` w) = D <-> E.x e. A E.y e. B D = C))
57	7, 56	sylan9bbr 541	. 2 |- ((A.x e. A A.y e. B C e. R /\ F Fn (A X. B)) -> (D e. ran F <-> E.x e. A E.y e. B D = C))
58	6, 57	mpdan 704	1 |- (A.x e. A A.y e. B C e. R -> (D e. ran F <-> E.x e. A E.y e. B D = C))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646  Vcvv 1811  <.cop 2411   X. cxp 3168  ran crn 3171   Fn wfn 3177  ` cfv 3182  (class class class)co 3963  {copab2 3964

This theorem is referenced by:  elrnoprab 4125  blrn 7841

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-9 965  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-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  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-ral 1649  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  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-rel 3185  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-f 3194  df-fv 3198  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080

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