Theorem ssrnres 5108

Description: Subset of the range of a restriction. (Contributed by NM, 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		inss2 3380	. . . . 5 ⊢ (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
2		rnss 4892	. . . . 5 ⊢ ((𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) → ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵))
3	1, 2	ax-mp 5	. . . 4 ⊢ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵)
4		rnxpss 5097	. . . 4 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
5	3, 4	sstri 3188	. . 3 ⊢ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵
6		eqss 3194	. . 3 ⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵 ∧ 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵))))
7	5, 6	mpbiran 942	. 2 ⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
8		ssid 3199	. . . . . . . 8 ⊢ 𝐴 ⊆ 𝐴
9		ssv 3201	. . . . . . . 8 ⊢ 𝐵 ⊆ V
10		xpss12 4766	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V))
11	8, 9, 10	mp2an 426	. . . . . . 7 ⊢ (𝐴 × 𝐵) ⊆ (𝐴 × V)
12		sslin 3385	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ⊆ (𝐴 × V) → (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶 ∩ (𝐴 × V)))
13	11, 12	ax-mp 5	. . . . . 6 ⊢ (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶 ∩ (𝐴 × V))
14		df-res 4671	. . . . . 6 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
15	13, 14	sseqtrri 3214	. . . . 5 ⊢ (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶 ↾ 𝐴)
16		rnss 4892	. . . . 5 ⊢ ((𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶 ↾ 𝐴) → ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶 ↾ 𝐴))
17	15, 16	ax-mp 5	. . . 4 ⊢ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶 ↾ 𝐴)
18		sstr 3187	. . . 4 ⊢ ((𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ∧ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶 ↾ 𝐴)) → 𝐵 ⊆ ran (𝐶 ↾ 𝐴))
19	17, 18	mpan2 425	. . 3 ⊢ (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) → 𝐵 ⊆ ran (𝐶 ↾ 𝐴))
20		ssel 3173	. . . . . . 7 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → 𝑦 ∈ ran (𝐶 ↾ 𝐴)))
21		vex 2763	. . . . . . . 8 ⊢ 𝑦 ∈ V
22	21	elrn2 4904	. . . . . . 7 ⊢ (𝑦 ∈ ran (𝐶 ↾ 𝐴) ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴))
23	20, 22	imbitrdi 161	. . . . . 6 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴)))
24	23	ancrd 326	. . . . 5 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → (∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵)))
25	21	elrn2 4904	. . . . . 6 ⊢ (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)))
26		elin 3342	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
27		opelxp 4689	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
28	27	anbi2i 457	. . . . . . . 8 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
29	21	opelres 4947	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑥 ∈ 𝐴))
30	29	anbi1i 458	. . . . . . . . 9 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵))
31		anass 401	. . . . . . . . 9 ⊢ (((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
32	30, 31	bitr2i 185	. . . . . . . 8 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵))
33	26, 28, 32	3bitri 206	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵))
34	33	exbii 1616	. . . . . 6 ⊢ (∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥(⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵))
35		19.41v 1914	. . . . . 6 ⊢ (∃𝑥(⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵) ↔ (∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵))
36	25, 34, 35	3bitri 206	. . . . 5 ⊢ (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ (∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵))
37	24, 36	imbitrrdi 162	. . . 4 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → 𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵))))
38	37	ssrdv 3185	. . 3 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
39	19, 38	impbii 126	. 2 ⊢ (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ 𝐵 ⊆ ran (𝐶 ↾ 𝐴))
40	7, 39	bitr2i 185	1 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1503   ∈ wcel 2164  Vcvv 2760   ∩ cin 3152   ⊆ wss 3153  ⟨cop 3621   × cxp 4657  ran crn 4660   ↾ cres 4661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-dm 4669  df-rn 4670  df-res 4671

This theorem is referenced by:  rninxp  5109