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Theorem mpompt 5829
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
mpompt.1 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
Assertion
Ref Expression
mpompt (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 mpompt
StepHypRef Expression
1 iunxpconst 4567 . . 3 𝑥𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)
2 mpteq1 3980 . . 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 mpompt.1 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
54mpomptx 5828 . 2 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
63, 5eqtr3i 2138 1 (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  {csn 3495  cop 3498   ciun 3781  cmpt 3957   × cxp 4505  cmpo 5742
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-iun 3783  df-opab 3958  df-mpt 3959  df-xp 4513  df-rel 4514  df-oprab 5744  df-mpo 5745
This theorem is referenced by:  fconstmpo  5832  fnovim  5845  fmpoco  6079  xpf1o  6704  txbas  12333  cnmpt1st  12363  cnmpt2nd  12364  cnmpt2c  12365  cnmpt2t  12368  txhmeo  12394  txswaphmeolem  12395
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