MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfmpt Structured version   Visualization version   GIF version

Theorem dfmpt 6553
Description: Alternate definition for the "maps to" notation df-mpt 4864 (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 6154 . 2 (𝑥𝐴𝐵) = 𝑥𝐴 ({𝑥} × {𝐵})
2 vex 3354 . . . . 5 𝑥 ∈ V
3 dfmpt.1 . . . . 5 𝐵 ∈ V
42, 3xpsn 6550 . . . 4 ({𝑥} × {𝐵}) = {⟨𝑥, 𝐵⟩}
54a1i 11 . . 3 (𝑥𝐴 → ({𝑥} × {𝐵}) = {⟨𝑥, 𝐵⟩})
65iuneq2i 4673 . 2 𝑥𝐴 ({𝑥} × {𝐵}) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}
71, 6eqtri 2793 1 (𝑥𝐴𝐵) = 𝑥𝐴 {⟨𝑥, 𝐵⟩}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wcel 2145  Vcvv 3351  {csn 4316  cop 4322   ciun 4654  cmpt 4863   × cxp 5247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038
This theorem is referenced by:  fnasrn  6554  funiun  6555  dfmpt2  7418
  Copyright terms: Public domain W3C validator