Theorem mpt2eq123i 5510

Description: An equality inference for the maps to notation. (Contributed by NM, 15-Jul-2013.)

Hypotheses

Ref	Expression
mpt2eq123i.1	⊢ A = 𝐷
mpt2eq123i.2	⊢ B = 𝐸
mpt2eq123i.3	⊢ 𝐶 = 𝐹

Assertion

Ref	Expression
mpt2eq123i	⊢ (x ∈ A, y ∈ B ↦ 𝐶) = (x ∈ 𝐷, y ∈ 𝐸 ↦ 𝐹)

Proof of Theorem mpt2eq123i

Step	Hyp	Ref	Expression
1		mpt2eq123i.1	. . . 4 ⊢ A = 𝐷
2	1	a1i 9	. . 3 ⊢ ( ⊤ → A = 𝐷)
3		mpt2eq123i.2	. . . 4 ⊢ B = 𝐸
4	3	a1i 9	. . 3 ⊢ ( ⊤ → B = 𝐸)
5		mpt2eq123i.3	. . . 4 ⊢ 𝐶 = 𝐹
6	5	a1i 9	. . 3 ⊢ ( ⊤ → 𝐶 = 𝐹)
7	2, 4, 6	mpt2eq123dv 5509	. 2 ⊢ ( ⊤ → (x ∈ A, y ∈ B ↦ 𝐶) = (x ∈ 𝐷, y ∈ 𝐸 ↦ 𝐹))
8	7	trud 1251	1 ⊢ (x ∈ A, y ∈ B ↦ 𝐶) = (x ∈ 𝐷, y ∈ 𝐸 ↦ 𝐹)

Syntax hints:   = wceq 1242   ⊤ wtru 1243   ↦ 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:  ofmres  5705