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Theorem mpompt 5945
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 4671 . . 3 𝑥𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)
2 mpteq1 4073 . . 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 5944 . 2 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
63, 5eqtr3i 2193 1 (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  {csn 3583  cop 3586   ciun 3873  cmpt 4050   × cxp 4609  cmpo 5855
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-iun 3875  df-opab 4051  df-mpt 4052  df-xp 4617  df-rel 4618  df-oprab 5857  df-mpo 5858
This theorem is referenced by:  fconstmpo  5948  fnovim  5961  fmpoco  6195  xpf1o  6822  txbas  13052  cnmpt1st  13082  cnmpt2nd  13083  cnmpt2c  13084  cnmpt2t  13087  txhmeo  13113  txswaphmeolem  13114
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