Theorem mpteq2i 4176

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 4175	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)

Syntax hints:   = wceq 1397   ∈ wcel 2202   ↦ cmpt 4150

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-ral 2515  df-opab 4151  df-mpt 4152

This theorem is referenced by:  frecsuc  6572  fodjuomni  7347  fodjumkv  7358  axcaucvg  8119  0tonninf  10701  1tonninf  10702  cbvsum  11920  cbvprod  12118  eirraplem  12337  znzrh2  14659  cnmpt12f  15009  fsumcncntop  15290  dvmptfsum  15448  dvef  15450  plyco  15482  plycj  15484  nninfsellemqall  16617  nninfomni  16621  nnnninfex  16624  exmidsbthr  16627