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Theorem mpt2mpt 6717
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 5146 . . 3 𝑥𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)
2 mpteq1 4707 . . 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 6716 . 2 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
63, 5eqtr3i 2645 1 (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  {csn 4155  cop 4161   ciun 4492  cmpt 4683   × cxp 5082  cmpt2 6617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-iun 4494  df-opab 4684  df-mpt 4685  df-xp 5090  df-rel 5091  df-oprab 6619  df-mpt2 6620
This theorem is referenced by:  fconstmpt2  6720  fnov  6733  fmpt2co  7220  xpf1o  8082  resfval2  16493  catcisolem  16696  xpccatid  16768  curf2ndf  16827  evlslem4  19448  mdetunilem9  20366  txbas  21310  cnmpt1st  21411  cnmpt2nd  21412  cnmpt2c  21413  cnmpt2t  21416  txhmeo  21546  txswaphmeolem  21547  ptuncnv  21550  ptunhmeo  21551  xpstopnlem1  21552  xkohmeo  21558  prdstmdd  21867  ucnimalem  22024  fmucndlem  22035  fsum2cn  22614  fimaproj  29724  curfv  33060  idfusubc0  41183  lmod1zr  41600
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