Theorem mpteq12i 4121

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 4115	. 2 ⊢ (⊤ → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
6	5	mptru 1373	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)

Syntax hints:   = wceq 1364  ⊤wtru 1365   ↦ cmpt 4094

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-ral 2480  df-opab 4095  df-mpt 4096

This theorem is referenced by:  offres  6192