Theorem mpteq2da 4178

Description: Slightly more general equality inference for the maps-to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.)

Hypotheses

Ref	Expression
mpteq2da.1	⊢ Ⅎ𝑥𝜑
mpteq2da.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)

Assertion

Ref	Expression
mpteq2da	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))

Proof of Theorem mpteq2da

Step	Hyp	Ref	Expression
1		eqid 2231	. . 3 ⊢ 𝐴 = 𝐴
2	1	ax-gen 1497	. 2 ⊢ ∀𝑥 𝐴 = 𝐴
3		mpteq2da.1	. . 3 ⊢ Ⅎ𝑥𝜑
4		mpteq2da.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)
5	4	ex 115	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 = 𝐶))
6	3, 5	ralrimi 2603	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶)
7		mpteq12f 4169	. 2 ⊢ ((∀𝑥 𝐴 = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
8	2, 6, 7	sylancr 414	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1395   = wceq 1397  Ⅎwnf 1508   ∈ wcel 2202  ∀wral 2510   ↦ 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:  mpteq2dva  4179  prodeq1f  12118  prodeq2  12123  gsumfzsnfd  13937