Theorem mpompt 7531

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) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)

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

Step	Hyp	Ref	Expression
1		iunxpconst 5732	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)
2	1	mpteq1i 5200	. 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶)
3		mpompt.1	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
4	3	mpomptx 7530	. 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)
5	2, 4	eqtr3i 2787	1 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  {csn 4587  ⟨cop 4593  ∪ ciun 4954   ↦ cmpt 5190   × cxp 5657   ∈ cmpo 7419

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-iun 4956  df-opab 5172  df-mpt 5191  df-xp 5665  df-rel 5666  df-oprab 7421  df-mpo 7422

This theorem is used by:  fconstmpo  7534  fnov  7548  fmpoco  8096  fimaproj  8137  curfv  8875  xpf1o  9141  resfval2  17988  idfusubc0  17994  catcisolem  18205  xpccatid  18282  curf2ndf  18341  evlslem4  22298  mdetunilem9  22848  txbas  23799  cnmpt1st  23900  cnmpt2nd  23901  cnmpt2c  23902  cnmpt2t  23905  txhmeo  24035  txswaphmeolem  24036  ptuncnv  24039  ptunhmeo  24040  xpstopnlem1  24041  xkohmeo  24047  prdstmdd  24356  ucnimalem  24511  fmucndlem  24522  fsum2cn  25105  conjga  33618  elrgspnlem2  33691  mplvrpmga  34063  aks6d1c2p1  42992  aks6d1c3  42997  aks6d1c4  42998  aks6d1c6lem2  43045  aks6d1c6lem4  43047  aks6d1c7lem1  43054  fmpocos  43111  lmod1zr  49431  2arymaptf  49590  iinfssclem1  49988  idfudiag1  50459