Theorem ssrnres 4937

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 3261	. . . . 5 ⊢ (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
2		rnss 4727	. . . . 5 ⊢ ((𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) → ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵))
3	1, 2	ax-mp 7	. . . 4 ⊢ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵)
4		rnxpss 4926	. . . 4 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
5	3, 4	sstri 3070	. . 3 ⊢ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵
6		eqss 3076	. . 3 ⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵 ∧ 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵))))
7	5, 6	mpbiran 905	. 2 ⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
8		ssid 3081	. . . . . . . 8 ⊢ 𝐴 ⊆ 𝐴
9		ssv 3083	. . . . . . . 8 ⊢ 𝐵 ⊆ V
10		xpss12 4604	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V))
11	8, 9, 10	mp2an 420	. . . . . . 7 ⊢ (𝐴 × 𝐵) ⊆ (𝐴 × V)
12		sslin 3266	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ⊆ (𝐴 × V) → (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶 ∩ (𝐴 × V)))
13	11, 12	ax-mp 7	. . . . . 6 ⊢ (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶 ∩ (𝐴 × V))
14		df-res 4509	. . . . . 6 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
15	13, 14	sseqtr4i 3096	. . . . 5 ⊢ (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶 ↾ 𝐴)
16		rnss 4727	. . . . 5 ⊢ ((𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶 ↾ 𝐴) → ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶 ↾ 𝐴))
17	15, 16	ax-mp 7	. . . 4 ⊢ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶 ↾ 𝐴)
18		sstr 3069	. . . 4 ⊢ ((𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ∧ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶 ↾ 𝐴)) → 𝐵 ⊆ ran (𝐶 ↾ 𝐴))
19	17, 18	mpan2 419	. . 3 ⊢ (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) → 𝐵 ⊆ ran (𝐶 ↾ 𝐴))
20		ssel 3055	. . . . . . 7 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → 𝑦 ∈ ran (𝐶 ↾ 𝐴)))
21		vex 2658	. . . . . . . 8 ⊢ 𝑦 ∈ V
22	21	elrn2 4739	. . . . . . 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 322	. . . . 5 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → (∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵)))
25	21	elrn2 4739	. . . . . 6 ⊢ (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)))
26		elin 3223	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
27		opelxp 4527	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
28	27	anbi2i 450	. . . . . . . 8 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
29	21	opelres 4780	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑥 ∈ 𝐴))
30	29	anbi1i 451	. . . . . . . . 9 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵))
31		anass 396	. . . . . . . . 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 1565	. . . . . 6 ⊢ (∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥(⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵))
35		19.41v 1854	. . . . . 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 3067	. . 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 1312  ∃wex 1449   ∈ wcel 1461  Vcvv 2655   ∩ cin 3034   ⊆ wss 3035  ⟨cop 3494   × cxp 4495  ran crn 4498   ↾ cres 4499

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-xp 4503  df-rel 4504  df-cnv 4505  df-dm 4507  df-rn 4508  df-res 4509

This theorem is referenced by:  rninxp  4938