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 5748	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)
2	1	mpteq1i 5244	. 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 2761	1 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   = wceq 1540  {csn 4628  ⟨cop 4634  ∪ ciun 4997   ↦ cmpt 5231   × cxp 5674   ∈ cmpo 7414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-iun 4999  df-opab 5211  df-mpt 5232  df-xp 5682  df-rel 5683  df-oprab 7416  df-mpo 7417

This theorem is referenced by:  fconstmpo  7528  fnov  7543  fmpoco  8086  fimaproj  8126  xpf1o  9145  resfval2  17850  idfusubc0  17856  catcisolem  18070  xpccatid  18150  curf2ndf  18210  evlslem4  21948  mdetunilem9  22442  txbas  23391  cnmpt1st  23492  cnmpt2nd  23493  cnmpt2c  23494  cnmpt2t  23497  txhmeo  23627  txswaphmeolem  23628  ptuncnv  23631  ptunhmeo  23632  xpstopnlem1  23633  xkohmeo  23639  prdstmdd  23948  ucnimalem  24105  fmucndlem  24116  fsum2cn  24709  curfv  36934  aks6d1c2p1  41425  fmpocos  41525  lmod1zr  47338  2arymaptf  47502