Theorem mpt2eq123dv 5587

Description: An equality deduction for the maps to notation. (Contributed by NM, 12-Sep-2011.)

Hypotheses

Ref	Expression
mpt2eq123dv.1	⊢ (𝜑 → 𝐴 = 𝐷)
mpt2eq123dv.2	⊢ (𝜑 → 𝐵 = 𝐸)
mpt2eq123dv.3	⊢ (𝜑 → 𝐶 = 𝐹)

Assertion

Ref	Expression
mpt2eq123dv	⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹))

Distinct variable groups:   𝜑,𝑥   𝜑,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem mpt2eq123dv

Step	Hyp	Ref	Expression
1		mpt2eq123dv.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐷)
2		mpt2eq123dv.2	. . 3 ⊢ (𝜑 → 𝐵 = 𝐸)
3	2	adantr 270	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐸)
4		mpt2eq123dv.3	. . 3 ⊢ (𝜑 → 𝐶 = 𝐹)
5	4	adantr 270	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 = 𝐹)
6	1, 3, 5	mpt2eq123dva 5586	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹))

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433   ↦ cmpt2 5534

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-oprab 5536  df-mpt2 5537

This theorem is referenced by:  mpt2eq123i  5588