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Theorem dfmpt 5368
Description: Alternate definition for the "maps to" notation df-mpt 3848 (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 5049 . 2 (𝑥𝐴𝐵) = 𝑥𝐴 ({𝑥} × {𝐵})
2 vex 2577 . . . . 5 𝑥 ∈ V
3 dfmpt.1 . . . . 5 𝐵 ∈ V
42, 3xpsn 5367 . . . 4 ({𝑥} × {𝐵}) = {⟨𝑥, 𝐵⟩}
54a1i 9 . . 3 (𝑥𝐴 → ({𝑥} × {𝐵}) = {⟨𝑥, 𝐵⟩})
65iuneq2i 3703 . 2 𝑥𝐴 ({𝑥} × {𝐵}) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}
71, 6eqtri 2076 1 (𝑥𝐴𝐵) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wcel 1409  Vcvv 2574  {csn 3403  cop 3406   ciun 3685  cmpt 3846   × cxp 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937
This theorem is referenced by:  fnasrn  5369  dfmpt2  5872
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