Theorem dfres3 5896

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 5601	. 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		eleq1 2826	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ 𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
3		vex 3436	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
4	3	biantru 530	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐵 ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ V))
5		vex 3436	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
6	5, 3	opelrn 5852	. . . . . . . . . . . 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 285	. . . . . . . . . 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 5612	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 × V) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ V)))
14		elxp 5612	. . . . . 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 575	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × V)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × ran 𝐴)))
17		elin 3903	. . . 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 4138	. 2 ⊢ (𝐴 ∩ (𝐵 × V)) = (𝐴 ∩ (𝐵 × ran 𝐴))
20	1, 19	eqtri 2766	1 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × ran 𝐴))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  Vcvv 3432   ∩ cin 3886  ⟨cop 4567   × cxp 5587  ran crn 5590   ↾ cres 5591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601

This theorem is referenced by:  brrestrict  34251  dfrel6  36482