Theorem ssrnres 4981

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 3297	. . . . 5 ⊢ (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
2		rnss 4769	. . . . 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 4970	. . . 4 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
5	3, 4	sstri 3106	. . 3 ⊢ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵
6		eqss 3112	. . 3 ⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵 ∧ 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵))))
7	5, 6	mpbiran 924	. 2 ⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
8		ssid 3117	. . . . . . . 8 ⊢ 𝐴 ⊆ 𝐴
9		ssv 3119	. . . . . . . 8 ⊢ 𝐵 ⊆ V
10		xpss12 4646	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V))
11	8, 9, 10	mp2an 422	. . . . . . 7 ⊢ (𝐴 × 𝐵) ⊆ (𝐴 × V)
12		sslin 3302	. . . . . . 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 4551	. . . . . 6 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
15	13, 14	sseqtrri 3132	. . . . 5 ⊢ (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶 ↾ 𝐴)
16		rnss 4769	. . . . 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 3105	. . . 4 ⊢ ((𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ∧ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶 ↾ 𝐴)) → 𝐵 ⊆ ran (𝐶 ↾ 𝐴))
19	17, 18	mpan2 421	. . 3 ⊢ (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) → 𝐵 ⊆ ran (𝐶 ↾ 𝐴))
20		ssel 3091	. . . . . . 7 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → 𝑦 ∈ ran (𝐶 ↾ 𝐴)))
21		vex 2689	. . . . . . . 8 ⊢ 𝑦 ∈ V
22	21	elrn2 4781	. . . . . . 7 ⊢ (𝑦 ∈ ran (𝐶 ↾ 𝐴) ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴))
23	20, 22	syl6ib 160	. . . . . 6 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴)))
24	23	ancrd 324	. . . . 5 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → (∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵)))
25	21	elrn2 4781	. . . . . 6 ⊢ (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)))
26		elin 3259	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
27		opelxp 4569	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
28	27	anbi2i 452	. . . . . . . 8 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
29	21	opelres 4824	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑥 ∈ 𝐴))
30	29	anbi1i 453	. . . . . . . . 9 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵))
31		anass 398	. . . . . . . . 9 ⊢ (((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
32	30, 31	bitr2i 184	. . . . . . . 8 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵))
33	26, 28, 32	3bitri 205	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵))
34	33	exbii 1584	. . . . . 6 ⊢ (∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥(⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵))
35		19.41v 1874	. . . . . 6 ⊢ (∃𝑥(⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵) ↔ (∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵))
36	25, 34, 35	3bitri 205	. . . . 5 ⊢ (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ (∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵))
37	24, 36	syl6ibr 161	. . . 4 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → 𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵))))
38	37	ssrdv 3103	. . 3 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
39	19, 38	impbii 125	. 2 ⊢ (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ 𝐵 ⊆ ran (𝐶 ↾ 𝐴))
40	7, 39	bitr2i 184	1 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480  Vcvv 2686   ∩ cin 3070   ⊆ wss 3071  ⟨cop 3530   × cxp 4537  ran crn 4540   ↾ cres 4541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551

This theorem is referenced by:  rninxp  4982