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Theorem elimasng 4755
Description: Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.)
Assertion
Ref Expression
elimasng ((𝐵𝑉𝐶𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))

Proof of Theorem elimasng
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sneq 3433 . . . . 5 (𝑦 = 𝐵 → {𝑦} = {𝐵})
21imaeq2d 4729 . . . 4 (𝑦 = 𝐵 → (𝐴 “ {𝑦}) = (𝐴 “ {𝐵}))
32eleq2d 2152 . . 3 (𝑦 = 𝐵 → (𝑧 ∈ (𝐴 “ {𝑦}) ↔ 𝑧 ∈ (𝐴 “ {𝐵})))
4 opeq1 3596 . . . 4 (𝑦 = 𝐵 → ⟨𝑦, 𝑧⟩ = ⟨𝐵, 𝑧⟩)
54eleq1d 2151 . . 3 (𝑦 = 𝐵 → (⟨𝑦, 𝑧⟩ ∈ 𝐴 ↔ ⟨𝐵, 𝑧⟩ ∈ 𝐴))
63, 5bibi12d 233 . 2 (𝑦 = 𝐵 → ((𝑧 ∈ (𝐴 “ {𝑦}) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴) ↔ (𝑧 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝑧⟩ ∈ 𝐴)))
7 eleq1 2145 . . 3 (𝑧 = 𝐶 → (𝑧 ∈ (𝐴 “ {𝐵}) ↔ 𝐶 ∈ (𝐴 “ {𝐵})))
8 opeq2 3597 . . . 4 (𝑧 = 𝐶 → ⟨𝐵, 𝑧⟩ = ⟨𝐵, 𝐶⟩)
98eleq1d 2151 . . 3 (𝑧 = 𝐶 → (⟨𝐵, 𝑧⟩ ∈ 𝐴 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))
107, 9bibi12d 233 . 2 (𝑧 = 𝐶 → ((𝑧 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝑧⟩ ∈ 𝐴) ↔ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)))
11 vex 2615 . . 3 𝑦 ∈ V
12 vex 2615 . . 3 𝑧 ∈ V
1311, 12elimasn 4754 . 2 (𝑧 ∈ (𝐴 “ {𝑦}) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴)
146, 10, 13vtocl2g 2673 1 ((𝐵𝑉𝐶𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  {csn 3422  cop 3425  cima 4404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-xp 4407  df-cnv 4409  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414
This theorem is referenced by:  eliniseg  4757  inimasn  4803  dffv3g  5249  fvimacnv  5359  funfvima3  5468  elecg  6260
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