Theorem mpteq1 4048

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

Assertion

Ref	Expression
mpteq1	⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem mpteq1

Step	Hyp	Ref	Expression
1		eqidd 2158	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐶 = 𝐶)
2	1	rgen 2510	. 2 ⊢ ∀𝑥 ∈ 𝐴 𝐶 = 𝐶
3		mpteq12 4047	. 2 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))
4	2, 3	mpan2 422	1 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))

Syntax hints:   → wi 4   = wceq 1335   ∈ wcel 2128  ∀wral 2435   ↦ cmpt 4025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-ral 2440  df-opab 4026  df-mpt 4027

This theorem is referenced by:  mpteq1d  4049  fmptap  5654  mpompt  5907  mpomptsx  6139  mpompts  6140  tposf12  6210  restco  12534  cnmpt1t  12645  cnmpt2t  12653