Theorem mpteq12i 4140

Description: An equality inference for the maps-to notation. (Contributed by Scott Fenton, 27-Oct-2010.) (Revised by Mario Carneiro, 16-Dec-2013.)

Hypotheses

Ref	Expression
mpteq12i.1	⊢ 𝐴 = 𝐶
mpteq12i.2	⊢ 𝐵 = 𝐷

Assertion

Ref	Expression
mpteq12i	⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)

Proof of Theorem mpteq12i

Step	Hyp	Ref	Expression
1		mpteq12i.1	. . . 4 ⊢ 𝐴 = 𝐶
2	1	a1i 9	. . 3 ⊢ (⊤ → 𝐴 = 𝐶)
3		mpteq12i.2	. . . 4 ⊢ 𝐵 = 𝐷
4	3	a1i 9	. . 3 ⊢ (⊤ → 𝐵 = 𝐷)
5	2, 4	mpteq12dv 4134	. 2 ⊢ (⊤ → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
6	5	mptru 1382	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)

Syntax hints:   = wceq 1373  ⊤wtru 1374   ↦ cmpt 4113

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-ral 2490  df-opab 4114  df-mpt 4115

This theorem is referenced by:  offres  6233