Theorem ssrnres 5059

Description: Subset of the range of a restriction. (Contributed by set.mm contributors, 16-Jan-2006.)

Assertion

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

Proof of Theorem ssrnres

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqss 3287	. 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 3476	. . . . 5 |- (C i^i (A X. B)) C_ (A X. B)
3		rnss 4959	. . . . 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 5	. . . 4 |- ran (C i^i (A X. B)) C_ ran (A X. B)
5		rnxpss 5053	. . . 4 |- ran (A X. B) C_ B
6	4, 5	sstri 3281	. . 3 |- ran (C i^i (A X. B)) C_ B
7	6	biantrur 492	. 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		ssv 3291	. . . . . . . 8 |- B C_ _V
9		xpss2 4857	. . . . . . . 8 |- (B C_ _V -> (A X. B) C_ (A X. _V))
10	8, 9	ax-mp 5	. . . . . . 7 |- (A X. B) C_ (A X. _V)
11		sslin 3481	. . . . . . 7 |- ((A X. B) C_ (A X. _V) -> (C i^i (A X. B)) C_ (C i^i (A X. _V)))
12	10, 11	ax-mp 5	. . . . . 6 |- (C i^i (A X. B)) C_ (C i^i (A X. _V))
13		df-res 4788	. . . . . 6 |- (C |` A) = (C i^i (A X. _V))
14	12, 13	sseqtr4i 3304	. . . . 5 |- (C i^i (A X. B)) C_ (C |` A)
15		rnss 4959	. . . . 5 |- ((C i^i (A X. B)) C_ (C |` A) -> ran (C i^i (A X. B)) C_ ran (C |` A))
16	14, 15	ax-mp 5	. . . 4 |- ran (C i^i (A X. B)) C_ ran (C |` A)
17		sstr 3280	. . . 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))
18	16, 17	mpan2 652	. . 3 |- (B C_ ran (C i^i (A X. B)) -> B C_ ran (C |` A))
19		ssel 3267	. . . . . . 7 |- (B C_ ran (C |` A) -> (y e. B -> y e. ran (C |` A)))
20		elrn2 4897	. . . . . . 7 |- (y e. ran (C |` A) <-> E.x<.x, y>. e. (C |` A))
21	19, 20	syl6ib 217	. . . . . 6 |- (B C_ ran (C |` A) -> (y e. B -> E.x<.x, y>. e. (C |` A)))
22	21	ancrd 537	. . . . 5 |- (B C_ ran (C |` A) -> (y e. B -> (E.x<.x, y>. e. (C |` A) /\ y e. B)))
23		elrn2 4897	. . . . . 6 |- (y e. ran (C i^i (A X. B)) <-> E.x<.x, y>. e. (C i^i (A X. B)))
24		elin 3219	. . . . . . . 8 |- (<.x, y>. e. (C i^i (A X. B)) <-> (<.x, y>. e. C /\ <.x, y>. e. (A X. B)))
25		opelxp 4811	. . . . . . . . 9 |- (<.x, y>. e. (A X. B) <-> (x e. A /\ y e. B))
26	25	anbi2i 675	. . . . . . . 8 |- ((<.x, y>. e. C /\ <.x, y>. e. (A X. B)) <-> (<.x, y>. e. C /\ (x e. A /\ y e. B)))
27		opelres 4950	. . . . . . . . . 10 |- (<.x, y>. e. (C |` A) <-> (<.x, y>. e. C /\ x e. A))
28	27	anbi1i 676	. . . . . . . . 9 |- ((<.x, y>. e. (C |` A) /\ y e. B) <-> ((<.x, y>. e. C /\ x e. A) /\ y e. B))
29		anass 630	. . . . . . . . 9 |- (((<.x, y>. e. C /\ x e. A) /\ y e. B) <-> (<.x, y>. e. C /\ (x e. A /\ y e. B)))
30	28, 29	bitr2i 241	. . . . . . . 8 |- ((<.x, y>. e. C /\ (x e. A /\ y e. B)) <-> (<.x, y>. e. (C |` A) /\ y e. B))
31	24, 26, 30	3bitri 262	. . . . . . 7 |- (<.x, y>. e. (C i^i (A X. B)) <-> (<.x, y>. e. (C |` A) /\ y e. B))
32	31	exbii 1582	. . . . . 6 |- (E.x<.x, y>. e. (C i^i (A X. B)) <-> E.x(<.x, y>. e. (C |` A) /\ y e. B))
33		19.41v 1901	. . . . . 6 |- (E.x(<.x, y>. e. (C |` A) /\ y e. B) <-> (E.x<.x, y>. e. (C |` A) /\ y e. B))
34	23, 32, 33	3bitri 262	. . . . 5 |- (y e. ran (C i^i (A X. B)) <-> (E.x<.x, y>. e. (C |` A) /\ y e. B))
35	22, 34	syl6ibr 218	. . . 4 |- (B C_ ran (C |` A) -> (y e. B -> y e. ran (C i^i (A X. B))))
36	35	ssrdv 3278	. . 3 |- (B C_ ran (C |` A) -> B C_ ran (C i^i (A X. B)))
37	18, 36	impbii 180	. 2 |- (B C_ ran (C i^i (A X. B)) <-> B C_ ran (C |` A))
38	1, 7, 37	3bitr2ri 265	1 |- (B C_ ran (C |` A) <-> ran (C i^i (A X. B)) = B)

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2859   i^i cin 3208   C_ wss 3257  <.cop 4561   X. cxp 4770  ran crn 4773   |` cres 4774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-ima 4727  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788

This theorem is referenced by:  rninxp  5060