Theorem mpteq1 4178

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 2232	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐶 = 𝐶)
2	1	rgen 2586	. 2 ⊢ ∀𝑥 ∈ 𝐴 𝐶 = 𝐶
3		mpteq12 4177	. 2 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))
4	2, 3	mpan2 425	1 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2202  ∀wral 2511   ↦ cmpt 4155

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-ral 2516  df-opab 4156  df-mpt 4157

This theorem is referenced by:  mpteq1d  4179  fmptap  5852  mpompt  6123  mpomptsx  6371  mpompts  6372  tposf12  6478  restco  14968  cnmpt1t  15079  cnmpt2t  15087