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Theorem dfres3 30902
 Description: Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
dfres3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × ran 𝐴))

Proof of Theorem dfres3
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-res 5050 . 2 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
2 eleq1 2676 . . . . . . . . . 10 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
3 vex 3176 . . . . . . . . . . . 12 𝑧 ∈ V
43biantru 525 . . . . . . . . . . 11 (𝑦𝐵 ↔ (𝑦𝐵𝑧 ∈ V))
5 vex 3176 . . . . . . . . . . . . 13 𝑦 ∈ V
65, 3opelrn 5278 . . . . . . . . . . . 12 (⟨𝑦, 𝑧⟩ ∈ 𝐴𝑧 ∈ ran 𝐴)
76biantrud 527 . . . . . . . . . . 11 (⟨𝑦, 𝑧⟩ ∈ 𝐴 → (𝑦𝐵 ↔ (𝑦𝐵𝑧 ∈ ran 𝐴)))
84, 7syl5bbr 273 . . . . . . . . . 10 (⟨𝑦, 𝑧⟩ ∈ 𝐴 → ((𝑦𝐵𝑧 ∈ V) ↔ (𝑦𝐵𝑧 ∈ ran 𝐴)))
92, 8syl6bi 242 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥𝐴 → ((𝑦𝐵𝑧 ∈ V) ↔ (𝑦𝐵𝑧 ∈ ran 𝐴))))
109com12 32 . . . . . . . 8 (𝑥𝐴 → (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦𝐵𝑧 ∈ V) ↔ (𝑦𝐵𝑧 ∈ ran 𝐴))))
1110pm5.32d 669 . . . . . . 7 (𝑥𝐴 → ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ V)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ ran 𝐴))))
12112exbidv 1839 . . . . . 6 (𝑥𝐴 → (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ V)) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ ran 𝐴))))
13 elxp 5055 . . . . . 6 (𝑥 ∈ (𝐵 × V) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ V)))
14 elxp 5055 . . . . . 6 (𝑥 ∈ (𝐵 × ran 𝐴) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ ran 𝐴)))
1512, 13, 143bitr4g 302 . . . . 5 (𝑥𝐴 → (𝑥 ∈ (𝐵 × V) ↔ 𝑥 ∈ (𝐵 × ran 𝐴)))
1615pm5.32i 667 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵 × V)) ↔ (𝑥𝐴𝑥 ∈ (𝐵 × ran 𝐴)))
17 elin 3758 . . . 4 (𝑥 ∈ (𝐴 ∩ (𝐵 × ran 𝐴)) ↔ (𝑥𝐴𝑥 ∈ (𝐵 × ran 𝐴)))
1816, 17bitr4i 266 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵 × V)) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵 × ran 𝐴)))
1918ineqri 3768 . 2 (𝐴 ∩ (𝐵 × V)) = (𝐴 ∩ (𝐵 × ran 𝐴))
201, 19eqtri 2632 1 (𝐴𝐵) = (𝐴 ∩ (𝐵 × ran 𝐴))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539  ⟨cop 4131   × cxp 5036  ran crn 5039   ↾ cres 5040 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050 This theorem is referenced by:  brrestrict  31226
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