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Theorem dfmpt 7135
Description: Alternate definition for the maps-to notation df-mpt 5223 (although it requires that 𝐵 be a set). (Contributed by NM, 24-Aug-2010.) (Revised by Mario Carneiro, 30-Dec-2016.)
Hypothesis
Ref Expression
dfmpt.1 𝐵 ∈ V
Assertion
Ref Expression
dfmpt (𝑥𝐴𝐵) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}

Proof of Theorem dfmpt
StepHypRef Expression
1 dfmpt3 6675 . 2 (𝑥𝐴𝐵) = 𝑥𝐴 ({𝑥} × {𝐵})
2 vex 3470 . . . . 5 𝑥 ∈ V
3 dfmpt.1 . . . . 5 𝐵 ∈ V
42, 3xpsn 7132 . . . 4 ({𝑥} × {𝐵}) = {⟨𝑥, 𝐵⟩}
54a1i 11 . . 3 (𝑥𝐴 → ({𝑥} × {𝐵}) = {⟨𝑥, 𝐵⟩})
65iuneq2i 5009 . 2 𝑥𝐴 ({𝑥} × {𝐵}) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}
71, 6eqtri 2752 1 (𝑥𝐴𝐵) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  Vcvv 3466  {csn 4621  cop 4627   ciun 4988  cmpt 5222   × cxp 5665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541
This theorem is referenced by:  fnasrn  7136  funiun  7138  dfmpo  8083
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