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Theorem elimasn 4692
 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 3768 . . 3 (𝑥 = 𝐶 → (𝐵𝐴𝑥𝐵𝐴𝐶))
3 elimasn.1 . . . 4 𝐵 ∈ V
4 imasng 4690 . . . 4 (𝐵 ∈ V → (𝐴 “ {𝐵}) = {𝑥𝐵𝐴𝑥})
53, 4ax-mp 7 . . 3 (𝐴 “ {𝐵}) = {𝑥𝐵𝐴𝑥}
61, 2, 5elab2 2690 . 2 (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)
7 df-br 3765 . 2 (𝐵𝐴𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
86, 7bitri 173 1 (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1243   ∈ wcel 1393  {cab 2026  Vcvv 2557  {csn 3375  ⟨cop 3378   class class class wbr 3764   “ cima 4348 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358 This theorem is referenced by:  elimasng  4693  dfco2  4820  dfco2a  4821  ressn  4858
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