Theorem mpteq12 5186

Description: An equality theorem for the maps-to notation. (Contributed by NM, 16-Dec-2013.)

Assertion

Ref	Expression
mpteq12	⊢ ((𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑥)

Proof of Theorem mpteq12

Step	Hyp	Ref	Expression
1		ax-5 1911	. 2 ⊢ (𝐴 = 𝐶 → ∀𝑥 𝐴 = 𝐶)
2		mpteq12f 5183	. 2 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
3	1, 2	sylan 580	1 ⊢ ((𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541  ∀wral 3051   ↦ cmpt 5179

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2184  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-opab 5161  df-mpt 5180

This theorem is referenced by:  mpteqb  6960  fmptcof  7075  mapxpen  9071  prodeq2w  15833  prdsdsval2  17404  prdsdsval3  17405  ablfac2  20020  mdetunilem9  22564  mdetmul  22567  xkocnv  23758  voliun  25511  itgeq1fOLD  25729  itgeq2  25735  iblcnlem  25746  bddiblnc  25799  esumeq2  34193  esumcvg  34243  dvtan  37871