Theorem elimasn 6090

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 6088, remove, and relabel elimasn1 6088 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 6089	. 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 2144  Vcvv 3450  {csn 4584  ⟨cop 4834   “ cima 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5657  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664

This theorem is used by:  dfco2  6249  dfco2a  6250  ressn  6291  funfvima3  7246  frxp  8145  frxp2  8168  frxp3  8175  marypha1lem  9432  gsum2dlem1  20193  gsum2dlem2  20194  gsum2d  20195  gsum2d2  20197  ovoliunlem1  25832  dmcuts  28186  cutsf  28187  iunsnima  33263  dfcnv2  33320  gsummpt2co  33659  gsummpt2d  33660  gsumfs2d  33672  funpartfun  36831  areaquad  44355  dffrege76  45076  frege97  45097  frege98  45098  frege109  45109  frege110  45110  frege131  45131  frege133  45133