Theorem mpteq12 5239

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 5235	. 2 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
3	1, 2	sylan 578	1 ⊢ ((𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 394  ∀wal 1537   = wceq 1539  ∀wral 3059   ↦ cmpt 5230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-opab 5210  df-mpt 5231

This theorem is referenced by:  mpteq1OLD  5241  mpteqb  7016  fmptcof  7129  mapxpen  9145  prodeq2w  15860  prdsdsval2  17434  prdsdsval3  17435  ablfac2  20000  mdetunilem9  22342  mdetmul  22345  xkocnv  23538  voliun  25303  itgeq1f  25521  itgeq2  25527  iblcnlem  25538  bddiblnc  25591  esumeq2  33332  esumcvg  33382  dvtan  36841