MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elimasngOLD Structured version   Visualization version   GIF version

Theorem elimasngOLD 6089
Description: Obsolete version of elimasng 6087 as of 16-Oct-2024. (Contributed by Raph Levien, 21-Oct-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
elimasngOLD ((𝐵𝑉𝐶𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))

Proof of Theorem elimasngOLD
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sneq 4634 . . . . 5 (𝑦 = 𝐵 → {𝑦} = {𝐵})
21imaeq2d 6058 . . . 4 (𝑦 = 𝐵 → (𝐴 “ {𝑦}) = (𝐴 “ {𝐵}))
32eleq2d 2811 . . 3 (𝑦 = 𝐵 → (𝑧 ∈ (𝐴 “ {𝑦}) ↔ 𝑧 ∈ (𝐴 “ {𝐵})))
4 opeq1 4869 . . . 4 (𝑦 = 𝐵 → ⟨𝑦, 𝑧⟩ = ⟨𝐵, 𝑧⟩)
54eleq1d 2810 . . 3 (𝑦 = 𝐵 → (⟨𝑦, 𝑧⟩ ∈ 𝐴 ↔ ⟨𝐵, 𝑧⟩ ∈ 𝐴))
63, 5bibi12d 344 . 2 (𝑦 = 𝐵 → ((𝑧 ∈ (𝐴 “ {𝑦}) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴) ↔ (𝑧 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝑧⟩ ∈ 𝐴)))
7 eleq1 2813 . . 3 (𝑧 = 𝐶 → (𝑧 ∈ (𝐴 “ {𝐵}) ↔ 𝐶 ∈ (𝐴 “ {𝐵})))
8 opeq2 4870 . . . 4 (𝑧 = 𝐶 → ⟨𝐵, 𝑧⟩ = ⟨𝐵, 𝐶⟩)
98eleq1d 2810 . . 3 (𝑧 = 𝐶 → (⟨𝐵, 𝑧⟩ ∈ 𝐴 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))
107, 9bibi12d 344 . 2 (𝑧 = 𝐶 → ((𝑧 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝑧⟩ ∈ 𝐴) ↔ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)))
11 vex 3467 . . 3 𝑦 ∈ V
12 vex 3467 . . 3 𝑧 ∈ V
1311, 12elimasn 6088 . 2 (𝑧 ∈ (𝐴 “ {𝑦}) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴)
146, 10, 13vtocl2g 3552 1 ((𝐵𝑉𝐶𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  {csn 4624  cop 4630  cima 5675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-xp 5678  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator