Theorem ssrnres 5572

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 3834	. . . . 5 ⊢ (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
2		rnss 5354	. . . . 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 5566	. . . 4 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
5	3, 4	sstri 3612	. . 3 ⊢ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵
6		eqss 3618	. . 3 ⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵 ∧ 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵))))
7	5, 6	mpbiran 953	. 2 ⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
8		ssid 3624	. . . . . . . 8 ⊢ 𝐴 ⊆ 𝐴
9		ssv 3625	. . . . . . . 8 ⊢ 𝐵 ⊆ V
10		xpss12 5225	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V))
11	8, 9, 10	mp2an 708	. . . . . . 7 ⊢ (𝐴 × 𝐵) ⊆ (𝐴 × V)
12		sslin 3839	. . . . . . 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 5126	. . . . . 6 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
15	13, 14	sseqtr4i 3638	. . . . 5 ⊢ (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶 ↾ 𝐴)
16		rnss 5354	. . . . 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 3611	. . . 4 ⊢ ((𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ∧ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶 ↾ 𝐴)) → 𝐵 ⊆ ran (𝐶 ↾ 𝐴))
19	17, 18	mpan2 707	. . 3 ⊢ (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) → 𝐵 ⊆ ran (𝐶 ↾ 𝐴))
20		ssel 3597	. . . . . . 7 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → 𝑦 ∈ ran (𝐶 ↾ 𝐴)))
21		vex 3203	. . . . . . . 8 ⊢ 𝑦 ∈ V
22	21	elrn2 5365	. . . . . . 7 ⊢ (𝑦 ∈ ran (𝐶 ↾ 𝐴) ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴))
23	20, 22	syl6ib 241	. . . . . 6 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴)))
24	23	ancrd 577	. . . . 5 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → (∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵)))
25	21	elrn2 5365	. . . . . 6 ⊢ (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)))
26		elin 3796	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
27		opelxp 5146	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
28	27	anbi2i 730	. . . . . . . 8 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
29	21	opelres 5401	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑥 ∈ 𝐴))
30	29	anbi1i 731	. . . . . . . . 9 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵))
31		anass 681	. . . . . . . . 9 ⊢ (((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
32	30, 31	bitr2i 265	. . . . . . . 8 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵))
33	26, 28, 32	3bitri 286	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵))
34	33	exbii 1774	. . . . . 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 286	. . . . 5 ⊢ (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ (∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ∧ 𝑦 ∈ 𝐵))
37	24, 36	syl6ibr 242	. . . 4 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → 𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵))))
38	37	ssrdv 3609	. . 3 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
39	19, 38	impbii 199	. 2 ⊢ (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ 𝐵 ⊆ ran (𝐶 ↾ 𝐴))
40	7, 39	bitr2i 265	1 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  ⟨cop 4183   × cxp 5112  ran crn 5115   ↾ cres 5116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by:  rninxp  5573