Theorem ssrnres 6209

Description: Two ways to express surjectivity of a restricted and corestricted binary relation (intersection of a binary relation with a Cartesian product): the LHS expresses inclusion in the range of the restricted relation, while the RHS expresses equality with the range of the restricted and corestricted relation. (Contributed by NM, 16-Jan-2006.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)

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 4259	. . . . 5 ⊢ (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
2	1	rnssi 5965	. . . 4 ⊢ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵)
3		rnxpss 6203	. . . 4 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
4	2, 3	sstri 4018	. . 3 ⊢ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵
5		eqss 4024	. . 3 ⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵 ∧ 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵))))
6	4, 5	mpbiran 708	. 2 ⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
7		inxpssres 5717	. . . . 5 ⊢ (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶 ↾ 𝐴)
8	7	rnssi 5965	. . . 4 ⊢ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶 ↾ 𝐴)
9		sstr 4017	. . . 4 ⊢ ((𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ∧ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶 ↾ 𝐴)) → 𝐵 ⊆ ran (𝐶 ↾ 𝐴))
10	8, 9	mpan2 690	. . 3 ⊢ (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) → 𝐵 ⊆ ran (𝐶 ↾ 𝐴))
11		ssel 4002	. . . . . . 7 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → 𝑦 ∈ ran (𝐶 ↾ 𝐴)))
12		vex 3492	. . . . . . . 8 ⊢ 𝑦 ∈ V
13	12	elrn2 5917	. . . . . . 7 ⊢ (𝑦 ∈ ran (𝐶 ↾ 𝐴) ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴))
14	11, 13	imbitrdi 251	. . . . . 6 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴)))
15	14	ancld 550	. . . . 5 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → (𝑦 ∈ 𝐵 ∧ ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴))))
16	12	elrn2 5917	. . . . . 6 ⊢ (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)))
17		opelinxp 5779	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐶))
18	12	opelresi 6017	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐶))
19	18	bianassc 642	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐶))
20	17, 19	bitr4i 278	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ (𝑦 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴)))
21	20	exbii 1846	. . . . . 6 ⊢ (∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥(𝑦 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴)))
22		19.42v 1953	. . . . . 6 ⊢ (∃𝑥(𝑦 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴)) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴)))
23	16, 21, 22	3bitri 297	. . . . 5 ⊢ (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴)))
24	15, 23	imbitrrdi 252	. . . 4 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → 𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵))))
25	24	ssrdv 4014	. . 3 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
26	10, 25	impbii 209	. 2 ⊢ (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ 𝐵 ⊆ ran (𝐶 ↾ 𝐴))
27	6, 26	bitr2i 276	1 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108   ∩ cin 3975   ⊆ wss 3976  ⟨cop 4654   × cxp 5698  ran crn 5701   ↾ cres 5702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712

This theorem is referenced by:  rninxp  6210