Theorem dfres3 5936

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 5630	. 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		eleq1 2827	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ 𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
3		vex 3435	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
4	3	biantru 534	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ V))
5		vex 3435	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
6	5, 3	opelrn 5885	. . . . . . . . . . . 12 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝐴 → 𝑧 ∈ ran 𝐴)
7	6	biantrud 536	. . . . . . . . . . 11 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝐴 → (𝑦 ∈ 𝐵 ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ran 𝐴)))
8	4, 7	bitr3id 286	. . . . . . . . . 10 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝐴 → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ V) ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ran 𝐴)))
9	2, 8	biimtrdi 254	. . . . . . . . 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 582	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ V)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ran 𝐴))))
12	11	2exbidv 1931	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ V)) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ran 𝐴))))
13		elxp 5641	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 × V) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ V)))
14		elxp 5641	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 × ran 𝐴) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ran 𝐴)))
15	12, 13, 14	3bitr4g 315	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ (𝐵 × V) ↔ 𝑥 ∈ (𝐵 × ran 𝐴)))
16	15	pm5.32i 579	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × V)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × ran 𝐴)))
17		elin 3899	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 × ran 𝐴)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × ran 𝐴)))
18	16, 17	bitr4i 279	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × V)) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵 × ran 𝐴)))
19	18	ineqri 4141	. 2 ⊢ (𝐴 ∩ (𝐵 × V)) = (𝐴 ∩ (𝐵 × ran 𝐴))
20	1, 19	eqtri 2762	1 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × ran 𝐴))

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  Vcvv 3431   ∩ cin 3882  ⟨cop 4561   × cxp 5616  ran crn 5619   ↾ cres 5620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624  df-cnv 5626  df-dm 5628  df-rn 5629  df-res 5630

This theorem is referenced by:  brrestrict  36177  dfrel6  38714