Theorem mpteq12i 5120

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 5112	. 2 ⊢ (⊤ → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
6	5	mptru 1549	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)

Syntax hints:   = wceq 1542  ⊤wtru 1543   ↦ cmpt 5107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-12 2178  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-opab 5090  df-mpt 5108

This theorem is referenced by:  partfun  6478  evlsval  20893  madufval  21381  cdj3lem3  30365  cdj3lem3b  30367  esumsnf  31594  esumrnmpt2  31598  measinb2  31753  eulerpart  31911  fiblem  31927  hoidmvlelem4  43662  smflimsup  43884