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Theorem dfmpt 5601
Description: Alternate definition for the maps-to notation df-mpt 3995 (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 5249 . 2 (𝑥𝐴𝐵) = 𝑥𝐴 ({𝑥} × {𝐵})
2 vex 2690 . . . . 5 𝑥 ∈ V
3 dfmpt.1 . . . . 5 𝐵 ∈ V
42, 3xpsn 5600 . . . 4 ({𝑥} × {𝐵}) = {⟨𝑥, 𝐵⟩}
54a1i 9 . . 3 (𝑥𝐴 → ({𝑥} × {𝐵}) = {⟨𝑥, 𝐵⟩})
65iuneq2i 3835 . 2 𝑥𝐴 ({𝑥} × {𝐵}) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}
71, 6eqtri 2161 1 (𝑥𝐴𝐵) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}
Colors of variables: wff set class
Syntax hints:   = wceq 1332  wcel 1481  Vcvv 2687  {csn 3528  cop 3531   ciun 3817  cmpt 3993   × cxp 4541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-v 2689  df-sbc 2911  df-csb 3005  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-id 4219  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134
This theorem is referenced by:  fnasrn  5602  dfmpo  6124
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