Theorem mpt2eq3ia 5570

Description: An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Hypothesis

Ref	Expression
mpt2eq3ia.1	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 = 𝐷)

Assertion

Ref	Expression
mpt2eq3ia	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)

Proof of Theorem mpt2eq3ia

Step	Hyp	Ref	Expression
1		mpt2eq3ia.1	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 = 𝐷)
2	1	3adant1 922	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 = 𝐷)
3	2	mpt2eq3dva 5569	. 2 ⊢ (⊤ → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷))
4	3	trud 1252	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ⊤wtru 1244   ∈ wcel 1393   ↦ cmpt2 5514

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-oprab 5516  df-mpt2 5517

This theorem is referenced by:  oprab2co  5839  genpdf  6606  dfioo2  8843