Theorem mpteq1 4216

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 2239	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐶 = 𝐶)
2	1	rgen 2603	. 2 ⊢ ∀𝑥 ∈ 𝐴 𝐶 = 𝐶
3		mpteq12 4215	. 2 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))
4	2, 3	mpan2 429	1 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∈ wcel 2209  ∀wral 2528   ↦ cmpt 4193

This proof depends on 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 proof 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 4194  df-mpt 4195

This theorem is used by:  mpteq1d  4217  fmptap  5908  mpompt  6183  mpomptsx  6437  mpompts  6438  tposf12  6544  indv  9304  gsumzfi  14248  gsummptfidmadd  14251  gsummptfidmsplit  14254  gsumconstcmn  14266  gsum2dfilem2  14272  gsumfsum  15032  restco  15400  cnmpt1t  15511  cnmpt2t  15519