Theorem mpteq2i 4213

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 4212	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)

Syntax hints:   = wceq 1402   ∈ wcel 2209   ↦ cmpt 4187

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-ral 2533  df-opab 4188  df-mpt 4189

This theorem is referenced by:  frecsuc  6668  fodjuomni  7479  fodjumkv  7490  axcaucvg  8257  0tonninf  10855  1tonninf  10856  cbvsum  12104  cbvprod  12303  eirraplem  12522  ballotfilemfc0  13210  ballotfilemfcc  13211  ballotfi  13260  znzrh2  14953  cnmpt12f  15310  fsumcncntop  15591  dvmptfsum  15749  dvef  15751  plyco  15783  plycj  15785  nninfsellemqall  16963  nninfomni  16967  nnnninfex  16970  exmidsbthr  16973