Theorem elimasng 4902

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 3533	. . . . 5 ⊢ (𝑦 = 𝐵 → {𝑦} = {𝐵})
2	1	imaeq2d 4876	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 “ {𝑦}) = (𝐴 “ {𝐵}))
3	2	eleq2d 2207	. . 3 ⊢ (𝑦 = 𝐵 → (𝑧 ∈ (𝐴 “ {𝑦}) ↔ 𝑧 ∈ (𝐴 “ {𝐵})))
4		opeq1 3700	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝑦, 𝑧⟩ = ⟨𝐵, 𝑧⟩)
5	4	eleq1d 2206	. . 3 ⊢ (𝑦 = 𝐵 → (⟨𝑦, 𝑧⟩ ∈ 𝐴 ↔ ⟨𝐵, 𝑧⟩ ∈ 𝐴))
6	3, 5	bibi12d 234	. 2 ⊢ (𝑦 = 𝐵 → ((𝑧 ∈ (𝐴 “ {𝑦}) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴) ↔ (𝑧 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝑧⟩ ∈ 𝐴)))
7		eleq1 2200	. . 3 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ (𝐴 “ {𝐵}) ↔ 𝐶 ∈ (𝐴 “ {𝐵})))
8		opeq2 3701	. . . 4 ⊢ (𝑧 = 𝐶 → ⟨𝐵, 𝑧⟩ = ⟨𝐵, 𝐶⟩)
9	8	eleq1d 2206	. . 3 ⊢ (𝑧 = 𝐶 → (⟨𝐵, 𝑧⟩ ∈ 𝐴 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))
10	7, 9	bibi12d 234	. 2 ⊢ (𝑧 = 𝐶 → ((𝑧 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝑧⟩ ∈ 𝐴) ↔ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)))
11		vex 2684	. . 3 ⊢ 𝑦 ∈ V
12		vex 2684	. . 3 ⊢ 𝑧 ∈ V
13	11, 12	elimasn 4901	. 2 ⊢ (𝑧 ∈ (𝐴 “ {𝑦}) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴)
14	6, 10, 13	vtocl2g 2745	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  {csn 3522  ⟨cop 3525   “ cima 4537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547

This theorem is referenced by:  eliniseg  4904  inimasn  4951  dffv3g  5410  fvimacnv  5528  funfvima3  5644  elecg  6460  imasnopn  12457