Theorem opkelopkabg 4245

Description: Kuratowski ordered pair membership in an abstraction of Kuratowski ordered pairs. (Contributed by SF, 12-Jan-2015.)

Hypotheses

Ref	Expression
opkelopkabg.1	⊢ A = {x ∣ ∃y∃z(x = ⟪y, z⟫ ∧ φ)}
opkelopkabg.2	⊢ (y = B → (φ ↔ ψ))
opkelopkabg.3	⊢ (z = C → (ψ ↔ χ))

Assertion

Ref	Expression
opkelopkabg	⊢ ((B ∈ V ∧ C ∈ W) → (⟪B, C⟫ ∈ A ↔ χ))

Distinct variable groups:   y,A,z   x,B,y,z   x,C,y,z   χ,z   φ,x   ψ,y   x,y,z

Allowed substitution hints:   φ(y,z)   ψ(x,z)   χ(x,y)   A(x)   V(x,y,z)   W(x,y,z)

Proof of Theorem opkelopkabg

Step	Hyp	Ref	Expression
1		opkex 4113	. . 3 ⊢ ⟪B, C⟫ ∈ V
2		eqeq1 2359	. . . . . 6 ⊢ (x = ⟪B, C⟫ → (x = ⟪y, z⟫ ↔ ⟪B, C⟫ = ⟪y, z⟫))
3		eqcom 2355	. . . . . 6 ⊢ (⟪B, C⟫ = ⟪y, z⟫ ↔ ⟪y, z⟫ = ⟪B, C⟫)
4	2, 3	syl6bb 252	. . . . 5 ⊢ (x = ⟪B, C⟫ → (x = ⟪y, z⟫ ↔ ⟪y, z⟫ = ⟪B, C⟫))
5	4	anbi1d 685	. . . 4 ⊢ (x = ⟪B, C⟫ → ((x = ⟪y, z⟫ ∧ φ) ↔ (⟪y, z⟫ = ⟪B, C⟫ ∧ φ)))
6	5	2exbidv 1628	. . 3 ⊢ (x = ⟪B, C⟫ → (∃y∃z(x = ⟪y, z⟫ ∧ φ) ↔ ∃y∃z(⟪y, z⟫ = ⟪B, C⟫ ∧ φ)))
7		opkelopkabg.1	. . 3 ⊢ A = {x ∣ ∃y∃z(x = ⟪y, z⟫ ∧ φ)}
8	1, 6, 7	elab2 2988	. 2 ⊢ (⟪B, C⟫ ∈ A ↔ ∃y∃z(⟪y, z⟫ = ⟪B, C⟫ ∧ φ))
9		elex 2867	. . 3 ⊢ (B ∈ V → B ∈ V)
10		elex 2867	. . 3 ⊢ (C ∈ W → C ∈ V)
11		vex 2862	. . . . . . . . . . 11 ⊢ y ∈ V
12		vex 2862	. . . . . . . . . . 11 ⊢ z ∈ V
13		opkthg 4131	. . . . . . . . . . 11 ⊢ ((y ∈ V ∧ z ∈ V ∧ C ∈ V) → (⟪y, z⟫ = ⟪B, C⟫ ↔ (y = B ∧ z = C)))
14	11, 12, 13	mp3an12 1267	. . . . . . . . . 10 ⊢ (C ∈ V → (⟪y, z⟫ = ⟪B, C⟫ ↔ (y = B ∧ z = C)))
15	14	adantl 452	. . . . . . . . 9 ⊢ ((B ∈ V ∧ C ∈ V) → (⟪y, z⟫ = ⟪B, C⟫ ↔ (y = B ∧ z = C)))
16	15	anbi1d 685	. . . . . . . 8 ⊢ ((B ∈ V ∧ C ∈ V) → ((⟪y, z⟫ = ⟪B, C⟫ ∧ φ) ↔ ((y = B ∧ z = C) ∧ φ)))
17		anass 630	. . . . . . . 8 ⊢ (((y = B ∧ z = C) ∧ φ) ↔ (y = B ∧ (z = C ∧ φ)))
18	16, 17	syl6bb 252	. . . . . . 7 ⊢ ((B ∈ V ∧ C ∈ V) → ((⟪y, z⟫ = ⟪B, C⟫ ∧ φ) ↔ (y = B ∧ (z = C ∧ φ))))
19	18	exbidv 1626	. . . . . 6 ⊢ ((B ∈ V ∧ C ∈ V) → (∃z(⟪y, z⟫ = ⟪B, C⟫ ∧ φ) ↔ ∃z(y = B ∧ (z = C ∧ φ))))
20		19.42v 1905	. . . . . 6 ⊢ (∃z(y = B ∧ (z = C ∧ φ)) ↔ (y = B ∧ ∃z(z = C ∧ φ)))
21	19, 20	syl6bb 252	. . . . 5 ⊢ ((B ∈ V ∧ C ∈ V) → (∃z(⟪y, z⟫ = ⟪B, C⟫ ∧ φ) ↔ (y = B ∧ ∃z(z = C ∧ φ))))
22	21	exbidv 1626	. . . 4 ⊢ ((B ∈ V ∧ C ∈ V) → (∃y∃z(⟪y, z⟫ = ⟪B, C⟫ ∧ φ) ↔ ∃y(y = B ∧ ∃z(z = C ∧ φ))))
23		opkelopkabg.2	. . . . . . . 8 ⊢ (y = B → (φ ↔ ψ))
24	23	anbi2d 684	. . . . . . 7 ⊢ (y = B → ((z = C ∧ φ) ↔ (z = C ∧ ψ)))
25	24	exbidv 1626	. . . . . 6 ⊢ (y = B → (∃z(z = C ∧ φ) ↔ ∃z(z = C ∧ ψ)))
26	25	ceqsexgv 2971	. . . . 5 ⊢ (B ∈ V → (∃y(y = B ∧ ∃z(z = C ∧ φ)) ↔ ∃z(z = C ∧ ψ)))
27	26	adantr 451	. . . 4 ⊢ ((B ∈ V ∧ C ∈ V) → (∃y(y = B ∧ ∃z(z = C ∧ φ)) ↔ ∃z(z = C ∧ ψ)))
28		opkelopkabg.3	. . . . . 6 ⊢ (z = C → (ψ ↔ χ))
29	28	ceqsexgv 2971	. . . . 5 ⊢ (C ∈ V → (∃z(z = C ∧ ψ) ↔ χ))
30	29	adantl 452	. . . 4 ⊢ ((B ∈ V ∧ C ∈ V) → (∃z(z = C ∧ ψ) ↔ χ))
31	22, 27, 30	3bitrd 270	. . 3 ⊢ ((B ∈ V ∧ C ∈ V) → (∃y∃z(⟪y, z⟫ = ⟪B, C⟫ ∧ φ) ↔ χ))
32	9, 10, 31	syl2an 463	. 2 ⊢ ((B ∈ V ∧ C ∈ W) → (∃y∃z(⟪y, z⟫ = ⟪B, C⟫ ∧ φ) ↔ χ))
33	8, 32	syl5bb 248	1 ⊢ ((B ∈ V ∧ C ∈ W) → (⟪B, C⟫ ∈ A ↔ χ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2859  ⟪copk 4057

This theorem was proved from 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  ax-nin 4078  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-opk 4058

This theorem is referenced by:  opkelopkab  4246  opkelxpkg  4247  opkelcnvkg  4249  opkelins2kg  4251  opkelins3kg  4252  opkelsikg  4264  opkelssetkg  4268  opkelidkg  4274  opklefing  4448  opkltfing  4449