Theorem ssrnres 4354

Description: Subset of the range of a restriction.

Assertion

Ref	Expression
ssrnres	|- (B C_ ran ( C |` A) <-> ran ( C i^i (A X. B)) = B)

Proof of Theorem ssrnres

Step	Hyp	Ref	Expression
1		eqss 2631	. 2 |- (ran ( C i^i (A X. B)) = B <-> (ran ( C i^i (A X. B)) C_ B /\ B C_ ran ( C i^i (A X. B))))
2		inss2 2813	. . . . 5 |- (C i^i (A X. B)) C_ (A X. B)
3		rnss 4189	. . . . 5 |- ((C i^i (A X. B)) C_ (A X. B) -> ran ( C i^i (A X. B)) C_ ran ( A X. B))
4	2, 3	ax-mp 7	. . . 4 |- ran ( C i^i (A X. B)) C_ ran ( A X. B)
5		rnxpss 4344	. . . 4 |- ran ( A X. B) C_ B
6	4, 5	sstri 2626	. . 3 |- ran ( C i^i (A X. B)) C_ B
7	6	biantrur 794	. 2 |- (B C_ ran ( C i^i (A X. B)) <-> (ran ( C i^i (A X. B)) C_ B /\ B C_ ran ( C i^i (A X. B))))
8		ssid 2634	. . . . . . . 8 |- A C_ A
9		ssv 2636	. . . . . . . 8 |- B C_ _V
10		xpss12 4089	. . . . . . . 8 |- ((A C_ A /\ B C_ _V) -> (A X. B) C_ (A X. _V))
11	8, 9, 10	mp2an 761	. . . . . . 7 |- (A X. B) C_ (A X. _V)
12		sslin 2819	. . . . . . 7 |- ((A X. B) C_ (A X. _V) -> (C i^i (A X. B)) C_ (C i^i (A X. _V)))
13	11, 12	ax-mp 7	. . . . . 6 |- (C i^i (A X. B)) C_ (C i^i (A X. _V))
14		df-res 4006	. . . . . 6 |- (C |` A) = (C i^i (A X. _V))
15	13, 14	sseqtr4i 2650	. . . . 5 |- (C i^i (A X. B)) C_ (C |` A)
16		rnss 4189	. . . . 5 |- ((C i^i (A X. B)) C_ (C |` A) -> ran ( C i^i (A X. B)) C_ ran ( C |` A))
17	15, 16	ax-mp 7	. . . 4 |- ran ( C i^i (A X. B)) C_ ran ( C |` A)
18		sstr 2625	. . . 4 |- ((B C_ ran ( C i^i (A X. B)) /\ ran ( C i^i (A X. B)) C_ ran ( C |` A)) -> B C_ ran ( C |` A))
19	17, 18	mpan2 760	. . 3 |- (B C_ ran ( C i^i (A X. B)) -> B C_ ran ( C |` A))
20		ssel 2615	. . . . . . 7 |- (B C_ ran ( C |` A) -> (y e. B -> y e. ran ( C |` A)))
21		visset 2295	. . . . . . . 8 |- y e. _V
22	21	elrn2 4196	. . . . . . 7 |- (y e. ran ( C |` A) <-> E.x<.x, y>. e. (C |` A))
23	20, 22	syl6ib 229	. . . . . 6 |- (B C_ ran ( C |` A) -> (y e. B -> E.x<.x, y>. e. (C |` A)))
24	23	ancrd 323	. . . . 5 |- (B C_ ran ( C |` A) -> (y e. B -> (E.x<.x, y>. e. (C |` A) /\ y e. B)))
25	21	elrn2 4196	. . . . . 6 |- (y e. ran ( C i^i (A X. B)) <-> E.x<.x, y>. e. (C i^i (A X. B)))
26		elin 2786	. . . . . . . 8 |- (<.x, y>. e. (C i^i (A X. B)) <-> (<.x, y>. e. C /\ <.x, y>. e. (A X. B)))
27	21	opelxp 4036	. . . . . . . . 9 |- (<.x, y>. e. (A X. B) <-> (x e. A /\ y e. B))
28	27	anbi2i 538	. . . . . . . 8 |- ((<.x, y>. e. C /\ <.x, y>. e. (A X. B)) <-> (<.x, y>. e. C /\ (x e. A /\ y e. B)))
29	21	opelres 4222	. . . . . . . . . 10 |- (<.x, y>. e. (C |` A) <-> (<.x, y>. e. C /\ x e. A))
30	29	anbi1i 539	. . . . . . . . 9 |- ((<.x, y>. e. (C |` A) /\ y e. B) <-> ((<.x, y>. e. C /\ x e. A) /\ y e. B))
31		anass 487	. . . . . . . . 9 |- (((<.x, y>. e. C /\ x e. A) /\ y e. B) <-> (<.x, y>. e. C /\ (x e. A /\ y e. B)))
32	30, 31	bitr2i 191	. . . . . . . 8 |- ((<.x, y>. e. C /\ (x e. A /\ y e. B)) <-> (<.x, y>. e. (C |` A) /\ y e. B))
33	26, 28, 32	3bitri 194	. . . . . . 7 |- (<.x, y>. e. (C i^i (A X. B)) <-> (<.x, y>. e. (C |` A) /\ y e. B))
34	33	exbii 1398	. . . . . 6 |- (E.x<.x, y>. e. (C i^i (A X. B)) <-> E.x(<.x, y>. e. (C |` A) /\ y e. B))
35		19.41v 1685	. . . . . 6 |- (E.x(<.x, y>. e. (C |` A) /\ y e. B) <-> (E.x<.x, y>. e. (C |` A) /\ y e. B))
36	25, 34, 35	3bitri 194	. . . . 5 |- (y e. ran ( C i^i (A X. B)) <-> (E.x<.x, y>. e. (C |` A) /\ y e. B))
37	24, 36	syl6ibr 230	. . . 4 |- (B C_ ran ( C |` A) -> (y e. B -> y e. ran ( C i^i (A X. B))))
38	37	ssrdv 2622	. . 3 |- (B C_ ran ( C |` A) -> B C_ ran ( C i^i (A X. B)))
39	19, 38	impbii 174	. 2 |- (B C_ ran ( C i^i (A X. B)) <-> B C_ ran ( C |` A))
40	1, 7, 39	3bitr2ri 197	1 |- (B C_ ran ( C |` A) <-> ran ( C i^i (A X. B)) = B)

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292   i^i cin 2592   C_ wss 2593  <.cop 3046   X. cxp 3984  ran crn 3987   |` cres 3988

This theorem is referenced by:  rninxp 4355  rninxpOLD 4356

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-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

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-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-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006

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