Theorem ssrnres 5060

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

Assertion

Ref	Expression
ssrnres	⊢ (B ⊆ ran (C ↾ A) ↔ ran (C ∩ (A × 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 3288	. 2 ⊢ (ran (C ∩ (A × B)) = B ↔ (ran (C ∩ (A × B)) ⊆ B ∧ B ⊆ ran (C ∩ (A × B))))
2		inss2 3477	. . . . 5 ⊢ (C ∩ (A × B)) ⊆ (A × B)
3		rnss 4960	. . . . 5 ⊢ ((C ∩ (A × B)) ⊆ (A × B) → ran (C ∩ (A × B)) ⊆ ran (A × B))
4	2, 3	ax-mp 5	. . . 4 ⊢ ran (C ∩ (A × B)) ⊆ ran (A × B)
5		rnxpss 5054	. . . 4 ⊢ ran (A × B) ⊆ B
6	4, 5	sstri 3282	. . 3 ⊢ ran (C ∩ (A × B)) ⊆ B
7	6	biantrur 492	. 2 ⊢ (B ⊆ ran (C ∩ (A × B)) ↔ (ran (C ∩ (A × B)) ⊆ B ∧ B ⊆ ran (C ∩ (A × B))))
8		ssv 3292	. . . . . . . 8 ⊢ B ⊆ V
9		xpss2 4858	. . . . . . . 8 ⊢ (B ⊆ V → (A × B) ⊆ (A × V))
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ (A × B) ⊆ (A × V)
11		sslin 3482	. . . . . . 7 ⊢ ((A × B) ⊆ (A × V) → (C ∩ (A × B)) ⊆ (C ∩ (A × V)))
12	10, 11	ax-mp 5	. . . . . 6 ⊢ (C ∩ (A × B)) ⊆ (C ∩ (A × V))
13		df-res 4789	. . . . . 6 ⊢ (C ↾ A) = (C ∩ (A × V))
14	12, 13	sseqtr4i 3305	. . . . 5 ⊢ (C ∩ (A × B)) ⊆ (C ↾ A)
15		rnss 4960	. . . . 5 ⊢ ((C ∩ (A × B)) ⊆ (C ↾ A) → ran (C ∩ (A × B)) ⊆ ran (C ↾ A))
16	14, 15	ax-mp 5	. . . 4 ⊢ ran (C ∩ (A × B)) ⊆ ran (C ↾ A)
17		sstr 3281	. . . 4 ⊢ ((B ⊆ ran (C ∩ (A × B)) ∧ ran (C ∩ (A × B)) ⊆ ran (C ↾ A)) → B ⊆ ran (C ↾ A))
18	16, 17	mpan2 652	. . 3 ⊢ (B ⊆ ran (C ∩ (A × B)) → B ⊆ ran (C ↾ A))
19		ssel 3268	. . . . . . 7 ⊢ (B ⊆ ran (C ↾ A) → (y ∈ B → y ∈ ran (C ↾ A)))
20		elrn2 4898	. . . . . . 7 ⊢ (y ∈ ran (C ↾ A) ↔ ∃x⟨x, y⟩ ∈ (C ↾ A))
21	19, 20	syl6ib 217	. . . . . 6 ⊢ (B ⊆ ran (C ↾ A) → (y ∈ B → ∃x⟨x, y⟩ ∈ (C ↾ A)))
22	21	ancrd 537	. . . . 5 ⊢ (B ⊆ ran (C ↾ A) → (y ∈ B → (∃x⟨x, y⟩ ∈ (C ↾ A) ∧ y ∈ B)))
23		elrn2 4898	. . . . . 6 ⊢ (y ∈ ran (C ∩ (A × B)) ↔ ∃x⟨x, y⟩ ∈ (C ∩ (A × B)))
24		elin 3220	. . . . . . . 8 ⊢ (⟨x, y⟩ ∈ (C ∩ (A × B)) ↔ (⟨x, y⟩ ∈ C ∧ ⟨x, y⟩ ∈ (A × B)))
25		opelxp 4812	. . . . . . . . 9 ⊢ (⟨x, y⟩ ∈ (A × B) ↔ (x ∈ A ∧ y ∈ B))
26	25	anbi2i 675	. . . . . . . 8 ⊢ ((⟨x, y⟩ ∈ C ∧ ⟨x, y⟩ ∈ (A × B)) ↔ (⟨x, y⟩ ∈ C ∧ (x ∈ A ∧ y ∈ B)))
27		opelres 4951	. . . . . . . . . 10 ⊢ (⟨x, y⟩ ∈ (C ↾ A) ↔ (⟨x, y⟩ ∈ C ∧ x ∈ A))
28	27	anbi1i 676	. . . . . . . . 9 ⊢ ((⟨x, y⟩ ∈ (C ↾ A) ∧ y ∈ B) ↔ ((⟨x, y⟩ ∈ C ∧ x ∈ A) ∧ y ∈ B))
29		anass 630	. . . . . . . . 9 ⊢ (((⟨x, y⟩ ∈ C ∧ x ∈ A) ∧ y ∈ B) ↔ (⟨x, y⟩ ∈ C ∧ (x ∈ A ∧ y ∈ B)))
30	28, 29	bitr2i 241	. . . . . . . 8 ⊢ ((⟨x, y⟩ ∈ C ∧ (x ∈ A ∧ y ∈ B)) ↔ (⟨x, y⟩ ∈ (C ↾ A) ∧ y ∈ B))
31	24, 26, 30	3bitri 262	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ (C ∩ (A × B)) ↔ (⟨x, y⟩ ∈ (C ↾ A) ∧ y ∈ B))
32	31	exbii 1582	. . . . . 6 ⊢ (∃x⟨x, y⟩ ∈ (C ∩ (A × B)) ↔ ∃x(⟨x, y⟩ ∈ (C ↾ A) ∧ y ∈ B))
33		19.41v 1901	. . . . . 6 ⊢ (∃x(⟨x, y⟩ ∈ (C ↾ A) ∧ y ∈ B) ↔ (∃x⟨x, y⟩ ∈ (C ↾ A) ∧ y ∈ B))
34	23, 32, 33	3bitri 262	. . . . 5 ⊢ (y ∈ ran (C ∩ (A × B)) ↔ (∃x⟨x, y⟩ ∈ (C ↾ A) ∧ y ∈ B))
35	22, 34	syl6ibr 218	. . . 4 ⊢ (B ⊆ ran (C ↾ A) → (y ∈ B → y ∈ ran (C ∩ (A × B))))
36	35	ssrdv 3279	. . 3 ⊢ (B ⊆ ran (C ↾ A) → B ⊆ ran (C ∩ (A × B)))
37	18, 36	impbii 180	. 2 ⊢ (B ⊆ ran (C ∩ (A × B)) ↔ B ⊆ ran (C ↾ A))
38	1, 7, 37	3bitr2ri 265	1 ⊢ (B ⊆ ran (C ↾ A) ↔ ran (C ∩ (A × B)) = B)

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∩ cin 3209   ⊆ wss 3258  ⟨cop 4562   × cxp 4771  ran crn 4774   ↾ cres 4775

This proof depends on 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof 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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-ima 4728  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789

This theorem is used by:  rninxp  5061