Theorem elimasng 5111

Description: Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.)

Assertion

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

Proof of Theorem elimasng

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

Step	Hyp	Ref	Expression
1		sneq 3684	. . . . 5 ⊢ (𝑦 = 𝐵 → {𝑦} = {𝐵})
2	1	imaeq2d 5082	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 “ {𝑦}) = (𝐴 “ {𝐵}))
3	2	eleq2d 2301	. . 3 ⊢ (𝑦 = 𝐵 → (𝑧 ∈ (𝐴 “ {𝑦}) ↔ 𝑧 ∈ (𝐴 “ {𝐵})))
4		opeq1 3867	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝑦, 𝑧⟩ = ⟨𝐵, 𝑧⟩)
5	4	eleq1d 2300	. . 3 ⊢ (𝑦 = 𝐵 → (⟨𝑦, 𝑧⟩ ∈ 𝐴 ↔ ⟨𝐵, 𝑧⟩ ∈ 𝐴))
6	3, 5	bibi12d 235	. 2 ⊢ (𝑦 = 𝐵 → ((𝑧 ∈ (𝐴 “ {𝑦}) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴) ↔ (𝑧 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝑧⟩ ∈ 𝐴)))
7		eleq1 2294	. . 3 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ (𝐴 “ {𝐵}) ↔ 𝐶 ∈ (𝐴 “ {𝐵})))
8		opeq2 3868	. . . 4 ⊢ (𝑧 = 𝐶 → ⟨𝐵, 𝑧⟩ = ⟨𝐵, 𝐶⟩)
9	8	eleq1d 2300	. . 3 ⊢ (𝑧 = 𝐶 → (⟨𝐵, 𝑧⟩ ∈ 𝐴 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))
10	7, 9	bibi12d 235	. 2 ⊢ (𝑧 = 𝐶 → ((𝑧 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝑧⟩ ∈ 𝐴) ↔ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)))
11		vex 2806	. . 3 ⊢ 𝑦 ∈ V
12		vex 2806	. . 3 ⊢ 𝑧 ∈ V
13	11, 12	elimasn 5110	. 2 ⊢ (𝑧 ∈ (𝐴 “ {𝑦}) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴)
14	6, 10, 13	vtocl2g 2869	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202  {csn 3673  ⟨cop 3676   “ cima 4734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744

This theorem is referenced by:  eliniseg  5113  inimasn  5161  dffv3g  5644  fvimacnv  5771  funfvima3  5898  elecg  6785  imasnopn  15093