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Theorem dfmpt 6903
 Description: Alternate definition for the maps-to notation df-mpt 5117 (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 6470 . 2 (𝑥𝐴𝐵) = 𝑥𝐴 ({𝑥} × {𝐵})
2 vex 3413 . . . . 5 𝑥 ∈ V
3 dfmpt.1 . . . . 5 𝐵 ∈ V
42, 3xpsn 6900 . . . 4 ({𝑥} × {𝐵}) = {⟨𝑥, 𝐵⟩}
54a1i 11 . . 3 (𝑥𝐴 → ({𝑥} × {𝐵}) = {⟨𝑥, 𝐵⟩})
65iuneq2i 4907 . 2 𝑥𝐴 ({𝑥} × {𝐵}) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}
71, 6eqtri 2781 1 (𝑥𝐴𝐵) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111  Vcvv 3409  {csn 4525  ⟨cop 4531  ∪ ciun 4886   ↦ cmpt 5116   × cxp 5526 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347 This theorem is referenced by:  fnasrn  6904  funiun  6906  dfmpo  7808
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