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	⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))

Proof of Theorem elimasngOLD

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

Step	Hyp	Ref	Expression
1		sneq 4638	. . . . 5 ⊢ (𝑦 = 𝐵 → {𝑦} = {𝐵})
2	1	imaeq2d 6059	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 “ {𝑦}) = (𝐴 “ {𝐵}))
3	2	eleq2d 2819	. . 3 ⊢ (𝑦 = 𝐵 → (𝑧 ∈ (𝐴 “ {𝑦}) ↔ 𝑧 ∈ (𝐴 “ {𝐵})))
4		opeq1 4873	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝑦, 𝑧⟩ = ⟨𝐵, 𝑧⟩)
5	4	eleq1d 2818	. . 3 ⊢ (𝑦 = 𝐵 → (⟨𝑦, 𝑧⟩ ∈ 𝐴 ↔ ⟨𝐵, 𝑧⟩ ∈ 𝐴))
6	3, 5	bibi12d 345	. 2 ⊢ (𝑦 = 𝐵 → ((𝑧 ∈ (𝐴 “ {𝑦}) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴) ↔ (𝑧 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝑧⟩ ∈ 𝐴)))
7		eleq1 2821	. . 3 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ (𝐴 “ {𝐵}) ↔ 𝐶 ∈ (𝐴 “ {𝐵})))
8		opeq2 4874	. . . 4 ⊢ (𝑧 = 𝐶 → ⟨𝐵, 𝑧⟩ = ⟨𝐵, 𝐶⟩)
9	8	eleq1d 2818	. . 3 ⊢ (𝑧 = 𝐶 → (⟨𝐵, 𝑧⟩ ∈ 𝐴 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))
10	7, 9	bibi12d 345	. 2 ⊢ (𝑧 = 𝐶 → ((𝑧 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝑧⟩ ∈ 𝐴) ↔ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)))
11		vex 3478	. . 3 ⊢ 𝑦 ∈ V
12		vex 3478	. . 3 ⊢ 𝑧 ∈ V
13	11, 12	elimasn 6088	. 2 ⊢ (𝑧 ∈ (𝐴 “ {𝑦}) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴)
14	6, 10, 13	vtocl2g 3562	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {csn 4628  ⟨cop 4634   “ cima 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689

This theorem is referenced by: (None)