Theorem ssrnres 3487

Description: Subset of the range of a restriction.

Assertion

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

Proof of Theorem ssrnres

Step	Hyp	Ref	Expression
1		eqss 2080	. 2 |- (ran ( C i^i (A X. B)) = B <-> (ran ( C i^i (A X. B)) (_ B /\ B (_ ran ( C i^i (A X. B))))
2		inss2 2234	. . . . 5 |- (C i^i (A X. B)) (_ (A X. B)
3		rnss 3348	. . . . 5 |- ((C i^i (A X. B)) (_ (A X. B) -> ran ( C i^i (A X. B)) (_ ran ( A X. B))
4	2, 3	ax-mp 7	. . . 4 |- ran ( C i^i (A X. B)) (_ ran ( A X. B)
5		rnxpss 3480	. . . 4 |- ran ( A X. B) (_ B
6	4, 5	sstri 2076	. . 3 |- ran ( C i^i (A X. B)) (_ B
7	6	biantrur 727	. 2 |- (B (_ ran ( C i^i (A X. B)) <-> (ran ( C i^i (A X. B)) (_ B /\ B (_ ran ( C i^i (A X. B))))
8		ssid 2083	. . . . . . . 8 |- A (_ A
9		ssv 2084	. . . . . . . 8 |- B (_ V
10		ssxp 3262	. . . . . . . 8 |- ((A (_ A /\ B (_ V) -> (A X. B) (_ (A X. V))
11	8, 9, 10	mp2an 699	. . . . . . 7 |- (A X. B) (_ (A X. V)
12		sslin 2238	. . . . . . 7 |- ((A X. B) (_ (A X. V) -> (C i^i (A X. B)) (_ (C i^i (A X. V)))
13	11, 12	ax-mp 7	. . . . . 6 |- (C i^i (A X. B)) (_ (C i^i (A X. V))
14		df-res 3196	. . . . . 6 |- (C |` A) = (C i^i (A X. V))
15	13, 14	sseqtr4 2097	. . . . 5 |- (C i^i (A X. B)) (_ (C |` A)
16		rnss 3348	. . . . 5 |- ((C i^i (A X. B)) (_ (C |` A) -> ran ( C i^i (A X. B)) (_ ran ( C |` A))
17	15, 16	ax-mp 7	. . . 4 |- ran ( C i^i (A X. B)) (_ ran ( C |` A)
18		sstr 2075	. . . 4 |- ((B (_ ran ( C i^i (A X. B)) /\ ran ( C i^i (A X. B)) (_ ran ( C |` A)) -> B (_ ran ( C |` A))
19	17, 18	mpan2 698	. . 3 |- (B (_ ran ( C i^i (A X. B)) -> B (_ ran ( C |` A))
20		ssel 2066	. . . . . . 7 |- (B (_ ran ( C |` A) -> (y e. B -> y e. ran ( C |` A)))
21		visset 1816	. . . . . . . 8 |- y e. V
22	21	elrn2 3355	. . . . . . 7 |- (y e. ran ( C |` A) <-> E.x<.x, y>. e. (C |` A))
23	20, 22	syl6ib 212	. . . . . 6 |- (B (_ ran ( C |` A) -> (y e. B -> E.x<.x, y>. e. (C |` A)))
24	23	ancrd 299	. . . . 5 |- (B (_ ran ( C |` A) -> (y e. B -> (E.x<.x, y>. e. (C |` A) /\ y e. B)))
25	21	elrn2 3355	. . . . . 6 |- (y e. ran ( C i^i (A X. B)) <-> E.x<.x, y>. e. (C i^i (A X. B)))
26		elin 2210	. . . . . . . 8 |- (<.x, y>. e. (C i^i (A X. B)) <-> (<.x, y>. e. C /\ <.x, y>. e. (A X. B)))
27	21	opelxp 3220	. . . . . . . . 9 |- (<.x, y>. e. (A X. B) <-> (x e. A /\ y e. B))
28	27	anbi2i 482	. . . . . . . 8 |- ((<.x, y>. e. C /\ <.x, y>. e. (A X. B)) <-> (<.x, y>. e. C /\ (x e. A /\ y e. B)))
29	21	opelres 3378	. . . . . . . . . 10 |- (<.x, y>. e. (C |` A) <-> (<.x, y>. e. C /\ x e. A))
30	29	anbi1i 483	. . . . . . . . 9 |- ((<.x, y>. e. (C |` A) /\ y e. B) <-> ((<.x, y>. e. C /\ x e. A) /\ y e. B))
31		anass 441	. . . . . . . . 9 |- (((<.x, y>. e. C /\ x e. A) /\ y e. B) <-> (<.x, y>. e. C /\ (x e. A /\ y e. B)))
32	30, 31	bitr2 174	. . . . . . . 8 |- ((<.x, y>. e. C /\ (x e. A /\ y e. B)) <-> (<.x, y>. e. (C |` A) /\ y e. B))
33	26, 28, 32	3bitr 177	. . . . . . 7 |- (<.x, y>. e. (C i^i (A X. B)) <-> (<.x, y>. e. (C |` A) /\ y e. B))
34	33	exbii 1053	. . . . . 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 1307	. . . . . 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	3bitr 177	. . . . 5 |- (y e. ran ( C i^i (A X. B)) <-> (E.x<.x, y>. e. (C |` A) /\ y e. B))
37	24, 36	syl6ibr 213	. . . 4 |- (B (_ ran ( C |` A) -> (y e. B -> y e. ran ( C i^i (A X. B))))
38	37	ssrdv 2073	. . 3 |- (B (_ ran ( C |` A) -> B (_ ran ( C i^i (A X. B)))
39	19, 38	impbi 157	. 2 |- (B (_ ran ( C i^i (A X. B)) <-> B (_ ran ( C |` A))
40	1, 7, 39	3bitr2r 180	1 |- (B (_ ran ( C |` A) <-> ran ( C i^i (A X. B)) = B)

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982  Vcvv 1814   i^i cin 2049   (_ wss 2050  <.cop 2415   X. cxp 3174  ran crn 3177   |` cres 3178

This theorem is referenced by:  rninxp 3488

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-9 967  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

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-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-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196

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