Theorem sikeq 4242

Description: Equality theorem for Kuratowski singleton image. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
sikeq	⊢ (A = B → SIk A = SIk B)

Proof of Theorem sikeq

Dummy variables x y z w t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2414	. . . . . . 7 ⊢ (A = B → (⟪w, t⟫ ∈ A ↔ ⟪w, t⟫ ∈ B))
2	1	3anbi3d 1258	. . . . . 6 ⊢ (A = B → ((y = {w} ∧ z = {t} ∧ ⟪w, t⟫ ∈ A) ↔ (y = {w} ∧ z = {t} ∧ ⟪w, t⟫ ∈ B)))
3	2	2exbidv 1628	. . . . 5 ⊢ (A = B → (∃w∃t(y = {w} ∧ z = {t} ∧ ⟪w, t⟫ ∈ A) ↔ ∃w∃t(y = {w} ∧ z = {t} ∧ ⟪w, t⟫ ∈ B)))
4	3	anbi2d 684	. . . 4 ⊢ (A = B → ((x = ⟪y, z⟫ ∧ ∃w∃t(y = {w} ∧ z = {t} ∧ ⟪w, t⟫ ∈ A)) ↔ (x = ⟪y, z⟫ ∧ ∃w∃t(y = {w} ∧ z = {t} ∧ ⟪w, t⟫ ∈ B))))
5	4	2exbidv 1628	. . 3 ⊢ (A = B → (∃y∃z(x = ⟪y, z⟫ ∧ ∃w∃t(y = {w} ∧ z = {t} ∧ ⟪w, t⟫ ∈ A)) ↔ ∃y∃z(x = ⟪y, z⟫ ∧ ∃w∃t(y = {w} ∧ z = {t} ∧ ⟪w, t⟫ ∈ B))))
6	5	abbidv 2468	. 2 ⊢ (A = B → {x ∣ ∃y∃z(x = ⟪y, z⟫ ∧ ∃w∃t(y = {w} ∧ z = {t} ∧ ⟪w, t⟫ ∈ A))} = {x ∣ ∃y∃z(x = ⟪y, z⟫ ∧ ∃w∃t(y = {w} ∧ z = {t} ∧ ⟪w, t⟫ ∈ B))})
7		df-sik 4193	. 2 ⊢ SIk A = {x ∣ ∃y∃z(x = ⟪y, z⟫ ∧ ∃w∃t(y = {w} ∧ z = {t} ∧ ⟪w, t⟫ ∈ A))}
8		df-sik 4193	. 2 ⊢ SIk B = {x ∣ ∃y∃z(x = ⟪y, z⟫ ∧ ∃w∃t(y = {w} ∧ z = {t} ∧ ⟪w, t⟫ ∈ B))}
9	6, 7, 8	3eqtr4g 2410	1 ⊢ (A = B → SIk A = SIk B)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  {csn 3738  ⟪copk 4058   SI_k csik 4182

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-sik 4193

This theorem is used by:  sikeqi  4243  sikeqd  4244  imagekeq  4245  sikexg  4297