Theorem mpteq2i 4104

Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Hypothesis

Ref	Expression
mpteq2i.1	⊢ 𝐵 = 𝐶

Assertion

Ref	Expression
mpteq2i	⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)

Proof of Theorem mpteq2i

Step	Hyp	Ref	Expression
1		mpteq2i.1	. . 3 ⊢ 𝐵 = 𝐶
2	1	a1i 9	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶)
3	2	mpteq2ia 4103	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)

Syntax hints:   = wceq 1363   ∈ wcel 2159   ↦ cmpt 4078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-ral 2472  df-opab 4079  df-mpt 4080

This theorem is referenced by:  frecsuc  6425  fodjuomni  7164  fodjumkv  7175  axcaucvg  7916  0tonninf  10456  1tonninf  10457  cbvsum  11385  cbvprod  11583  eirraplem  11801  cnmpt12f  14169  fsumcncntop  14439  dvef  14571  nninfsellemqall  15148  nninfomni  15152  exmidsbthr  15155