Theorem mpteq12i 5254

Description: An equality inference for the maps-to notation. (Contributed by Scott Fenton, 27-Oct-2010.) (Revised by Mario Carneiro, 16-Dec-2013.)

Hypotheses

Ref	Expression
mpteq12i.1	⊢ 𝐴 = 𝐶
mpteq12i.2	⊢ 𝐵 = 𝐷

Assertion

Ref	Expression
mpteq12i	⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)

Proof of Theorem mpteq12i

Step	Hyp	Ref	Expression
1		mpteq12i.1	. . . 4 ⊢ 𝐴 = 𝐶
2	1	a1i 11	. . 3 ⊢ (⊤ → 𝐴 = 𝐶)
3		mpteq12i.2	. . . 4 ⊢ 𝐵 = 𝐷
4	3	a1i 11	. . 3 ⊢ (⊤ → 𝐵 = 𝐷)
5	2, 4	mpteq12dv 5239	. 2 ⊢ (⊤ → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
6	5	mptru 1548	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)

Syntax hints:   = wceq 1541  ⊤wtru 1542   ↦ cmpt 5231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-opab 5211  df-mpt 5232

This theorem is referenced by:  partfun  6697  evlsval  21655  madufval  22146  cdj3lem3  31729  cdj3lem3b  31731  esumsnf  33131  esumrnmpt2  33135  measinb2  33290  eulerpart  33450  fiblem  33466  hoidmvlelem4  45393  smflimsup  45623