Theorem elimasn 6088

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 6086, remove, and relabel elimasn1 6086 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 6087	. 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 2145  Vcvv 3451  {csn 4584  ⟨cop 4590   “ cima 5654

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 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664

This theorem is used by:  dfco2  6246  dfco2a  6247  ressn  6288  funfvima3  7242  frxp  8138  frxp2  8161  frxp3  8168  marypha1lem  9425  gsum2dlem1  20184  gsum2dlem2  20185  gsum2d  20186  gsum2d2  20188  ovoliunlem1  25823  dmcuts  28177  cutsf  28178  iunsnima  33212  dfcnv2  33269  gsummpt2co  33609  gsummpt2d  33610  gsumfs2d  33622  funpartfun  36707  areaquad  44217  dffrege76  44938  frege97  44959  frege98  44960  frege109  44971  frege110  44972  frege131  44993  frege133  44995