New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  opkelopkabg GIF version

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 yz(x = ⟪y, z φ)}
opkelopkabg.2 (y = B → (φψ))
opkelopkabg.3 (z = C → (ψχ))
Assertion
Ref Expression
opkelopkabg ((B V C W) → (⟪B, C Aχ))
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
StepHypRef 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⟫)
42, 3syl6bb 252 . . . . 5 (x = ⟪B, C⟫ → (x = ⟪y, z⟫ ↔ ⟪y, z⟫ = ⟪B, C⟫))
54anbi1d 685 . . . 4 (x = ⟪B, C⟫ → ((x = ⟪y, z φ) ↔ (⟪y, z⟫ = ⟪B, C φ)))
652exbidv 1628 . . 3 (x = ⟪B, C⟫ → (yz(x = ⟪y, z φ) ↔ yz(⟪y, z⟫ = ⟪B, C φ)))
7 opkelopkabg.1 . . 3 A = {x yz(x = ⟪y, z φ)}
81, 6, 7elab2 2988 . 2 (⟪B, C Ayz(⟪y, z⟫ = ⟪B, C φ))
9 elex 2867 . . 3 (B VB V)
10 elex 2867 . . 3 (C WC 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)))
1411, 12, 13mp3an12 1267 . . . . . . . . . 10 (C V → (⟪y, z⟫ = ⟪B, C⟫ ↔ (y = B z = C)))
1514adantl 452 . . . . . . . . 9 ((B V C V) → (⟪y, z⟫ = ⟪B, C⟫ ↔ (y = B z = C)))
1615anbi1d 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)))
1816, 17syl6bb 252 . . . . . . 7 ((B V C V) → ((⟪y, z⟫ = ⟪B, C φ) ↔ (y = B (z = C φ))))
1918exbidv 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)))
2119, 20syl6bb 252 . . . . 5 ((B V C V) → (z(⟪y, z⟫ = ⟪B, C φ) ↔ (y = B z(z = C φ))))
2221exbidv 1626 . . . 4 ((B V C V) → (yz(⟪y, z⟫ = ⟪B, C φ) ↔ y(y = B z(z = C φ))))
23 opkelopkabg.2 . . . . . . . 8 (y = B → (φψ))
2423anbi2d 684 . . . . . . 7 (y = B → ((z = C φ) ↔ (z = C ψ)))
2524exbidv 1626 . . . . . 6 (y = B → (z(z = C φ) ↔ z(z = C ψ)))
2625ceqsexgv 2971 . . . . 5 (B V → (y(y = B z(z = C φ)) ↔ z(z = C ψ)))
2726adantr 451 . . . 4 ((B V C V) → (y(y = B z(z = C φ)) ↔ z(z = C ψ)))
28 opkelopkabg.3 . . . . . 6 (z = C → (ψχ))
2928ceqsexgv 2971 . . . . 5 (C V → (z(z = C ψ) ↔ χ))
3029adantl 452 . . . 4 ((B V C V) → (z(z = C ψ) ↔ χ))
3122, 27, 303bitrd 270 . . 3 ((B V C V) → (yz(⟪y, z⟫ = ⟪B, C φ) ↔ χ))
329, 10, 31syl2an 463 . 2 ((B V C W) → (yz(⟪y, z⟫ = ⟪B, C φ) ↔ χ))
338, 32syl5bb 248 1 ((B V C W) → (⟪B, C Aχ))
 Colors of variables: wff setvar class 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-1 5  ax-2 6  ax-3 7  ax-mp 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
 Copyright terms: Public domain W3C validator