Theorem elimasn 6094

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 6092, remove, and relabel elimasn1 6092 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

Step	Hyp	Ref	Expression
1		elimasn.1	. 2 ⊢ 𝐵 ∈ V
2		elimasn.2	. 2 ⊢ 𝐶 ∈ V
3		elimasng 6093	. 2 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))
4	1, 2, 3	mp2an 705	1 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2146  Vcvv 3457  {csn 4591  ⟨cop 4597   “ cima 5666

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676

This theorem is used by:  dfco2  6248  dfco2a  6249  ressn  6290  funfvima3  7241  frxp  8128  frxp2  8146  frxp3  8153  marypha1lem  9400  gsum2dlem1  20086  gsum2dlem2  20087  gsum2d  20088  gsum2d2  20090  ovoliunlem1  25714  dmcuts  28037  cutsf  28038  iunsnima  33036  dfcnv2  33093  gsummpt2co  33434  gsummpt2d  33435  gsumfs2d  33447  funpartfun  36474  areaquad  44003  dffrege76  44725  frege97  44746  frege98  44747  frege109  44758  frege110  44759  frege131  44780  frege133  44782