Theorem elimasn1 6078

Description: Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Use df-br 5103 and shorten proof. (Revised by BJ, 16-Oct-2024.)

Hypotheses

Ref	Expression
elimasn1.1	⊢ 𝐵 ∈ V
elimasn1.2	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
elimasn1	⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)

Proof of Theorem elimasn1

Step	Hyp	Ref	Expression
1		elimasn1.1	. 2 ⊢ 𝐵 ∈ V
2		elimasn1.2	. 2 ⊢ 𝐶 ∈ V
3		elimasng1 6077	. 2 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
4	1, 2, 3	mp2an 705	1 ⊢ (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2145  Vcvv 3450  {csn 4583   class class class wbr 5102   “ cima 5650

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 2732  ax-sep 5248  ax-pr 5390

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 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5653  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660

This theorem is used by: (None)