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Theorem elimasn 6045
Description: Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by BJ, 16-Oct-2024.) TODO: replace existing usages by usages of elimasn1 6043, remove, and relabel elimasn1 6043 to "elimasn".
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
StepHypRef Expression
1 elimasn.1 . 2 𝐵 ∈ V
2 elimasn.2 . 2 𝐶 ∈ V
3 elimasng 6044 . 2 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))
41, 2, 3mp2an 691 1 (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2107  Vcvv 3447  {csn 4590  cop 4596  cima 5640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650
This theorem is referenced by:  elimasngOLD  6046  dfco2  6201  dfco2a  6202  ressn  6241  funfvima3  7190  frxp  8062  frxp2  8080  frxp3  8087  marypha1lem  9377  gsum2dlem1  19755  gsum2dlem2  19756  gsum2d  19757  gsum2d2  19759  ovoliunlem1  24889  dmscut  27179  scutf  27180  iunsnima  31590  dfcnv2  31645  gsummpt2co  31946  gsummpt2d  31947  funpartfun  34581  areaquad  41597  dffrege76  42303  frege97  42324  frege98  42325  frege109  42336  frege110  42337  frege131  42358  frege133  42360
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