Theorem elimasng1 6097

Description: Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.) Revise to use df-br 5107 and to prove elimasn1 6098 from it. (Revised by BJ, 16-Oct-2024.)

Assertion

Ref	Expression
elimasng1	⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))

Proof of Theorem elimasng1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 490	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 𝐶 ∈ 𝑊)
2		imasng 6094	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐴 “ {𝐵}) = {𝑥 ∣ 𝐵𝐴𝑥})
3	2	adantr 486	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐴 “ {𝐵}) = {𝑥 ∣ 𝐵𝐴𝑥})
4		simpr 490	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 = 𝐶) → 𝑥 = 𝐶)
5	4	breq2d 5118	. 2 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 = 𝐶) → (𝐵𝐴𝑥 ↔ 𝐵𝐴𝐶))
6	1, 3, 5	elabd2 3627	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  {csn 4588   class class class wbr 5106   “ cima 5658

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668

This theorem is used by:  elimasn1  6098  elimasng  6099  elimasni  6101  elinisegg  6103