Theorem mpoeq123dv 6093

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

Hypotheses

Ref	Expression
mpoeq123dv.1	⊢ (𝜑 → 𝐴 = 𝐷)
mpoeq123dv.2	⊢ (𝜑 → 𝐵 = 𝐸)
mpoeq123dv.3	⊢ (𝜑 → 𝐶 = 𝐹)

Assertion

Ref	Expression
mpoeq123dv	⊢ (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 mpoeq123dv

Step	Hyp	Ref	Expression
1		mpoeq123dv.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐷)
2		mpoeq123dv.2	. . 3 ⊢ (𝜑 → 𝐵 = 𝐸)
3	2	adantr 276	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐸)
4		mpoeq123dv.3	. . 3 ⊢ (𝜑 → 𝐶 = 𝐹)
5	4	adantr 276	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 = 𝐹)
6	1, 3, 5	mpoeq123dva 6092	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2202   ∈ cmpo 6030

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-oprab 6032  df-mpo 6033

This theorem is referenced by:  mpoeq123i  6094  prdsex  13415  prdsval  13419  plusffvalg  13508  grpsubfvalg  13691  grpsubpropdg  13750  mulgfvalg  13771  mulgpropdg  13814  dvrfvald  14211  scaffvalg  14385  psrval  14745  blfvalps  15179  clwwlknonmpo  16352