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Theorem dfres3 5433
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 5155 . 2 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
2 eleq1 2718 . . . . . . . . . 10 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
3 vex 3234 . . . . . . . . . . . 12 𝑧 ∈ V
43biantru 525 . . . . . . . . . . 11 (𝑦𝐵 ↔ (𝑦𝐵𝑧 ∈ V))
5 vex 3234 . . . . . . . . . . . . 13 𝑦 ∈ V
65, 3opelrn 5389 . . . . . . . . . . . 12 (⟨𝑦, 𝑧⟩ ∈ 𝐴𝑧 ∈ ran 𝐴)
76biantrud 527 . . . . . . . . . . 11 (⟨𝑦, 𝑧⟩ ∈ 𝐴 → (𝑦𝐵 ↔ (𝑦𝐵𝑧 ∈ ran 𝐴)))
84, 7syl5bbr 274 . . . . . . . . . 10 (⟨𝑦, 𝑧⟩ ∈ 𝐴 → ((𝑦𝐵𝑧 ∈ V) ↔ (𝑦𝐵𝑧 ∈ ran 𝐴)))
92, 8syl6bi 243 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥𝐴 → ((𝑦𝐵𝑧 ∈ V) ↔ (𝑦𝐵𝑧 ∈ ran 𝐴))))
109com12 32 . . . . . . . 8 (𝑥𝐴 → (𝑥 = ⟨𝑦, 𝑧⟩ → ((𝑦𝐵𝑧 ∈ V) ↔ (𝑦𝐵𝑧 ∈ ran 𝐴))))
1110pm5.32d 672 . . . . . . 7 (𝑥𝐴 → ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ V)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ ran 𝐴))))
12112exbidv 1892 . . . . . 6 (𝑥𝐴 → (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ V)) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ ran 𝐴))))
13 elxp 5165 . . . . . 6 (𝑥 ∈ (𝐵 × V) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ V)))
14 elxp 5165 . . . . . 6 (𝑥 ∈ (𝐵 × ran 𝐴) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ ran 𝐴)))
1512, 13, 143bitr4g 303 . . . . 5 (𝑥𝐴 → (𝑥 ∈ (𝐵 × V) ↔ 𝑥 ∈ (𝐵 × ran 𝐴)))
1615pm5.32i 670 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵 × V)) ↔ (𝑥𝐴𝑥 ∈ (𝐵 × ran 𝐴)))
17 elin 3829 . . . 4 (𝑥 ∈ (𝐴 ∩ (𝐵 × ran 𝐴)) ↔ (𝑥𝐴𝑥 ∈ (𝐵 × ran 𝐴)))
1816, 17bitr4i 267 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵 × V)) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵 × ran 𝐴)))
1918ineqri 3839 . 2 (𝐴 ∩ (𝐵 × V)) = (𝐴 ∩ (𝐵 × ran 𝐴))
201, 19eqtri 2673 1 (𝐴𝐵) = (𝐴 ∩ (𝐵 × ran 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231  cin 3606  cop 4216   × cxp 5141  ran crn 5144  cres 5145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155
This theorem is referenced by:  brrestrict  32181  dfrel6  34255
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