Theorem dfres3 5958

Description: Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
dfres3	⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × ran 𝐴))

Proof of Theorem dfres3

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-res 5653	. 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		eleq1 2817	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ 𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
3		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
4	3	biantru 529	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ V))
5		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
6	5, 3	opelrn 5910	. . . . . . . . . . . 12 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝐴 → 𝑧 ∈ ran 𝐴)
7	6	biantrud 531	. . . . . . . . . . 11 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝐴 → (𝑦 ∈ 𝐵 ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ran 𝐴)))
8	4, 7	bitr3id 285	. . . . . . . . . 10 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝐴 → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ V) ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ran 𝐴)))
9	2, 8	biimtrdi 253	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ 𝐴 → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ V) ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ran 𝐴))))
10	9	com12 32	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ V) ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ran 𝐴))))
11	10	pm5.32d 577	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ V)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ran 𝐴))))
12	11	2exbidv 1924	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ V)) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ran 𝐴))))
13		elxp 5664	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 × V) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ V)))
14		elxp 5664	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 × ran 𝐴) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ran 𝐴)))
15	12, 13, 14	3bitr4g 314	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ (𝐵 × V) ↔ 𝑥 ∈ (𝐵 × ran 𝐴)))
16	15	pm5.32i 574	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × V)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × ran 𝐴)))
17		elin 3933	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 × ran 𝐴)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × ran 𝐴)))
18	16, 17	bitr4i 278	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × V)) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵 × ran 𝐴)))
19	18	ineqri 4178	. 2 ⊢ (𝐴 ∩ (𝐵 × V)) = (𝐴 ∩ (𝐵 × ran 𝐴))
20	1, 19	eqtri 2753	1 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × ran 𝐴))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3450   ∩ cin 3916  ⟨cop 4598   × cxp 5639  ran crn 5642   ↾ cres 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653

This theorem is referenced by:  brrestrict  35944  dfrel6  38336