Theorem dfres3 5986

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 5688	. 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		eleq1 2821	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ 𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
3		vex 3478	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
4	3	biantru 530	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ V))
5		vex 3478	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
6	5, 3	opelrn 5942	. . . . . . . . . . . 12 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝐴 → 𝑧 ∈ ran 𝐴)
7	6	biantrud 532	. . . . . . . . . . 11 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝐴 → (𝑦 ∈ 𝐵 ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ran 𝐴)))
8	4, 7	bitr3id 284	. . . . . . . . . 10 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝐴 → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ V) ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ran 𝐴)))
9	2, 8	syl6bi 252	. . . . . . . . 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 1927	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ V)) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ran 𝐴))))
13		elxp 5699	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 × V) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ V)))
14		elxp 5699	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 × ran 𝐴) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ran 𝐴)))
15	12, 13, 14	3bitr4g 313	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ (𝐵 × V) ↔ 𝑥 ∈ (𝐵 × ran 𝐴)))
16	15	pm5.32i 575	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × V)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × ran 𝐴)))
17		elin 3964	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ (𝐵 × ran 𝐴)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × ran 𝐴)))
18	16, 17	bitr4i 277	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × V)) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵 × ran 𝐴)))
19	18	ineqri 4204	. 2 ⊢ (𝐴 ∩ (𝐵 × V)) = (𝐴 ∩ (𝐵 × ran 𝐴))
20	1, 19	eqtri 2760	1 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × ran 𝐴))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  Vcvv 3474   ∩ cin 3947  ⟨cop 4634   × cxp 5674  ran crn 5677   ↾ cres 5678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688

This theorem is referenced by:  brrestrict  35213  dfrel6  37519