Theorem mpoeq123dv 5931

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 5930	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148   ∈ cmpo 5871

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-oprab 5873  df-mpo 5874

This theorem is referenced by:  mpoeq123i  5932  plusffvalg  12673  grpsubfvalg  12808  grpsubpropdg  12863  mulgfvalg  12874  mulgpropdg  12913  dvrfvald  13127  blfvalps  13552