Theorem mpompt 7525

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 5735	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)
2	1	mpteq1i 5206	. 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶)
3		mpompt.1	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
4	3	mpomptx 7524	. 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)
5	2, 4	eqtr3i 2794	1 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   = wceq 1567  {csn 4594  ⟨cop 4600  ∪ ciun 4960   ↦ cmpt 5196   × cxp 5660   ∈ cmpo 7413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-iun 4962  df-opab 5178  df-mpt 5197  df-xp 5668  df-rel 5669  df-oprab 7415  df-mpo 7416

This theorem is referenced by:  fconstmpo  7528  fnov  7542  fmpoco  8090  fimaproj  8131  xpf1o  9127  resfval2  17950  idfusubc0  17956  catcisolem  18167  xpccatid  18244  curf2ndf  18303  evlslem4  22196  mdetunilem9  22746  txbas  23693  cnmpt1st  23794  cnmpt2nd  23795  cnmpt2c  23796  cnmpt2t  23799  txhmeo  23929  txswaphmeolem  23930  ptuncnv  23933  ptunhmeo  23934  xpstopnlem1  23935  xkohmeo  23941  prdstmdd  24250  ucnimalem  24405  fmucndlem  24416  fsum2cn  24999  conjga  33431  elrgspnlem2  33504  mplvrpmga  33880  curfv  38139  aks6d1c2p1  42775  aks6d1c3  42780  aks6d1c4  42781  aks6d1c6lem2  42828  aks6d1c6lem4  42830  aks6d1c7lem1  42837  fmpocos  42894  lmod1zr  49158  2arymaptf  49317  iinfssclem1  49717  idfudiag1  50188