Theorem mpt2eq123dv 5509

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 261	. 2 ⊢ ((φ ∧ x ∈ A) → B = 𝐸)
4		mpt2eq123dv.3	. . 3 ⊢ (φ → 𝐶 = 𝐹)
5	4	adantr 261	. 2 ⊢ ((φ ∧ (x ∈ A ∧ y ∈ B)) → 𝐶 = 𝐹)
6	1, 3, 5	mpt2eq123dva 5508	1 ⊢ (φ → (x ∈ A, y ∈ B ↦ 𝐶) = (x ∈ 𝐷, y ∈ 𝐸 ↦ 𝐹))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390   ↦ cmpt2 5457

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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-oprab 5459  df-mpt2 5460

This theorem is referenced by:  mpt2eq123i  5510