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Theorem dfres3 31624
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 5116 . 2 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
2 eleq1 2687 . . . . . . . . . 10 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
3 vex 3198 . . . . . . . . . . . 12 𝑧 ∈ V
43biantru 526 . . . . . . . . . . 11 (𝑦𝐵 ↔ (𝑦𝐵𝑧 ∈ V))
5 vex 3198 . . . . . . . . . . . . 13 𝑦 ∈ V
65, 3opelrn 5346 . . . . . . . . . . . 12 (⟨𝑦, 𝑧⟩ ∈ 𝐴𝑧 ∈ ran 𝐴)
76biantrud 528 . . . . . . . . . . 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 670 . . . . . . 7 (𝑥𝐴 → ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ V)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ ran 𝐴))))
12112exbidv 1850 . . . . . 6 (𝑥𝐴 → (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ V)) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ ran 𝐴))))
13 elxp 5121 . . . . . 6 (𝑥 ∈ (𝐵 × V) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐵𝑧 ∈ V)))
14 elxp 5121 . . . . . 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 668 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵 × V)) ↔ (𝑥𝐴𝑥 ∈ (𝐵 × ran 𝐴)))
17 elin 3788 . . . 4 (𝑥 ∈ (𝐴 ∩ (𝐵 × ran 𝐴)) ↔ (𝑥𝐴𝑥 ∈ (𝐵 × ran 𝐴)))
1816, 17bitr4i 267 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵 × V)) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵 × ran 𝐴)))
1918ineqri 3798 . 2 (𝐴 ∩ (𝐵 × V)) = (𝐴 ∩ (𝐵 × ran 𝐴))
201, 19eqtri 2642 1 (𝐴𝐵) = (𝐴 ∩ (𝐵 × ran 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1481  wex 1702  wcel 1988  Vcvv 3195  cin 3566  cop 4174   × cxp 5102  ran crn 5105  cres 5106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-xp 5110  df-cnv 5112  df-dm 5114  df-rn 5115  df-res 5116
This theorem is referenced by:  brrestrict  32031
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