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Theorem mpt2mpt 6794
Description: Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 17-Dec-2013.) (Revised by Mario Carneiro, 29-Dec-2014.)
Hypothesis
Ref Expression
mpt2mpt.1 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
Assertion
Ref Expression
mpt2mpt (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑦,𝐵,𝑧   𝑥,𝐶,𝑦   𝑧,𝐷   𝑥,𝐵
Allowed substitution hints:   𝐶(𝑧)   𝐷(𝑥,𝑦)

Proof of Theorem mpt2mpt
StepHypRef Expression
1 iunxpconst 5209 . . 3 𝑥𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)
2 mpteq1 4770 . . 3 ( 𝑥𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵) → (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶))
31, 2ax-mp 5 . 2 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶)
4 mpt2mpt.1 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
54mpt2mptx 6793 . 2 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
63, 5eqtr3i 2675 1 (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  {csn 4210  cop 4216   ciun 4552  cmpt 4762   × cxp 5141  cmpt2 6692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-iun 4554  df-opab 4746  df-mpt 4763  df-xp 5149  df-rel 5150  df-oprab 6694  df-mpt2 6695
This theorem is referenced by:  fconstmpt2  6797  fnov  6810  fmpt2co  7305  xpf1o  8163  resfval2  16600  catcisolem  16803  xpccatid  16875  curf2ndf  16934  evlslem4  19556  mdetunilem9  20474  txbas  21418  cnmpt1st  21519  cnmpt2nd  21520  cnmpt2c  21521  cnmpt2t  21524  txhmeo  21654  txswaphmeolem  21655  ptuncnv  21658  ptunhmeo  21659  xpstopnlem1  21660  xkohmeo  21666  prdstmdd  21974  ucnimalem  22131  fmucndlem  22142  fsum2cn  22721  fimaproj  30028  curfv  33519  idfusubc0  42190  lmod1zr  42607
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