Theorem ssrnres 5053

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 3348	. . . . 5 ⊢ (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
2		rnss 4841	. . . . 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 5042	. . . 4 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
5	3, 4	sstri 3156	. . 3 ⊢ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵
6		eqss 3162	. . 3 ⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵 ∧ 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵))))
7	5, 6	mpbiran 935	. 2 ⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
8		ssid 3167	. . . . . . . 8 ⊢ 𝐴 ⊆ 𝐴
9		ssv 3169	. . . . . . . 8 ⊢ 𝐵 ⊆ V
10		xpss12 4718	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V))
11	8, 9, 10	mp2an 424	. . . . . . 7 ⊢ (𝐴 × 𝐵) ⊆ (𝐴 × V)
12		sslin 3353	. . . . . . 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 4623	. . . . . 6 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
15	13, 14	sseqtrri 3182	. . . . 5 ⊢ (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶 ↾ 𝐴)
16		rnss 4841	. . . . 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 3155	. . . 4 ⊢ ((𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ∧ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶 ↾ 𝐴)) → 𝐵 ⊆ ran (𝐶 ↾ 𝐴))
19	17, 18	mpan2 423	. . 3 ⊢ (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) → 𝐵 ⊆ ran (𝐶 ↾ 𝐴))
20		ssel 3141	. . . . . . 7 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → 𝑦 ∈ ran (𝐶 ↾ 𝐴)))
21		vex 2733	. . . . . . . 8 ⊢ 𝑦 ∈ V
22	21	elrn2 4853	. . . . . . 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 4853	. . . . . 6 ⊢ (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)))
26		elin 3310	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
27		opelxp 4641	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
28	27	anbi2i 454	. . . . . . . 8 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
29	21	opelres 4896	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑥 ∈ 𝐴))
30	29	anbi1i 455	. . . . . . . . 9 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵))
31		anass 399	. . . . . . . . 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 1598	. . . . . 6 ⊢ (∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥(⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵))
35		19.41v 1895	. . . . . 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 3153	. . 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 1348  ∃wex 1485   ∈ wcel 2141  Vcvv 2730   ∩ cin 3120   ⊆ wss 3121  ⟨cop 3586   × cxp 4609  ran crn 4612   ↾ cres 4613

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623

This theorem is referenced by:  rninxp  5054