Theorem mpteq12 5156

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 1910	. 2 ⊢ (𝐴 = 𝐶 → ∀𝑥 𝐴 = 𝐶)
2		mpteq12f 5152	. 2 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
3	1, 2	sylan 582	1 ⊢ ((𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1534   = wceq 1536  ∀wral 3141   ↦ cmpt 5149

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 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-ral 3146  df-opab 5132  df-mpt 5150

This theorem is referenced by:  mpteq1  5157  mpteqb  6790  fmptcof  6895  mapxpen  8686  prodeq2w  15269  prdsdsval2  16760  prdsdsval3  16761  ablfac2  19214  mdetunilem9  21232  mdetmul  21235  xkocnv  22425  voliun  24158  itgeq1f  24375  itgeq2  24381  iblcnlem  24392  esumeq2  31299  esumcvg  31349  dvtan  34946  bddiblnc  34966