Theorem ssrnres 6021

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 4194	. . . . 5 ⊢ (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
2	1	rnssi 5796	. . . 4 ⊢ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐴 × 𝐵)
3		rnxpss 6015	. . . 4 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵
4	2, 3	sstri 3964	. . 3 ⊢ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵
5		eqss 3970	. . 3 ⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ 𝐵 ∧ 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵))))
6	4, 5	mpbiran 707	. 2 ⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
7		inxpssres 5558	. . . . 5 ⊢ (𝐶 ∩ (𝐴 × 𝐵)) ⊆ (𝐶 ↾ 𝐴)
8	7	rnssi 5796	. . . 4 ⊢ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶 ↾ 𝐴)
9		sstr 3963	. . . 4 ⊢ ((𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ∧ ran (𝐶 ∩ (𝐴 × 𝐵)) ⊆ ran (𝐶 ↾ 𝐴)) → 𝐵 ⊆ ran (𝐶 ↾ 𝐴))
10	8, 9	mpan2 689	. . 3 ⊢ (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) → 𝐵 ⊆ ran (𝐶 ↾ 𝐴))
11		ssel 3949	. . . . . . 7 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → 𝑦 ∈ ran (𝐶 ↾ 𝐴)))
12		vex 3489	. . . . . . . 8 ⊢ 𝑦 ∈ V
13	12	elrn2 5807	. . . . . . 7 ⊢ (𝑦 ∈ ran (𝐶 ↾ 𝐴) ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴))
14	11, 13	syl6ib 253	. . . . . 6 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴)))
15	14	ancld 553	. . . . 5 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → (𝑦 ∈ 𝐵 ∧ ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴))))
16	12	elrn2 5807	. . . . . 6 ⊢ (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)))
17		opelinxp 5617	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐶))
18	12	opelresi 5847	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐶))
19	18	bianassc 641	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐶))
20	17, 19	bitr4i 280	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ (𝑦 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴)))
21	20	exbii 1848	. . . . . 6 ⊢ (∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥(𝑦 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴)))
22		19.42v 1954	. . . . . 6 ⊢ (∃𝑥(𝑦 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴)) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴)))
23	16, 21, 22	3bitri 299	. . . . 5 ⊢ (𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥⟨𝑥, 𝑦⟩ ∈ (𝐶 ↾ 𝐴)))
24	15, 23	syl6ibr 254	. . . 4 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → (𝑦 ∈ 𝐵 → 𝑦 ∈ ran (𝐶 ∩ (𝐴 × 𝐵))))
25	24	ssrdv 3961	. . 3 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) → 𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)))
26	10, 25	impbii 211	. 2 ⊢ (𝐵 ⊆ ran (𝐶 ∩ (𝐴 × 𝐵)) ↔ 𝐵 ⊆ ran (𝐶 ↾ 𝐴))
27	6, 26	bitr2i 278	1 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ∩ cin 3923   ⊆ wss 3924  ⟨cop 4559   × cxp 5539  ran crn 5542   ↾ cres 5543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5189  ax-nul 5196  ax-pr 5316

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3488  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-nul 4280  df-if 4454  df-sn 4554  df-pr 4556  df-op 4560  df-br 5053  df-opab 5115  df-xp 5547  df-rel 5548  df-cnv 5549  df-dm 5551  df-rn 5552  df-res 5553

This theorem is referenced by:  rninxp  6022