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Theorem elimasn 4763
Description: Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypotheses
Ref Expression
elimasn.1 𝐵 ∈ V
elimasn.2 𝐶 ∈ V
Assertion
Ref Expression
elimasn (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)

Proof of Theorem elimasn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elimasn.2 . . 3 𝐶 ∈ V
2 breq2 3824 . . 3 (𝑥 = 𝐶 → (𝐵𝐴𝑥𝐵𝐴𝐶))
3 elimasn.1 . . . 4 𝐵 ∈ V
4 imasng 4761 . . . 4 (𝐵 ∈ V → (𝐴 “ {𝐵}) = {𝑥𝐵𝐴𝑥})
53, 4ax-mp 7 . . 3 (𝐴 “ {𝐵}) = {𝑥𝐵𝐴𝑥}
61, 2, 5elab2 2754 . 2 (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)
7 df-br 3821 . 2 (𝐵𝐴𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
86, 7bitri 182 1 (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
Colors of variables: wff set class
Syntax hints:  wb 103   = wceq 1287  wcel 1436  {cab 2071  Vcvv 2615  {csn 3431  cop 3434   class class class wbr 3820  cima 4413
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3931  ax-pow 3983  ax-pr 4009
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3821  df-opab 3875  df-xp 4416  df-cnv 4418  df-dm 4420  df-rn 4421  df-res 4422  df-ima 4423
This theorem is referenced by:  elimasng  4764  dfco2  4893  dfco2a  4894  ressn  4934
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