Theorem elimasng1 6080

Description: Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.) Revise to use df-br 5106 and to prove elimasn1 6081 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 489	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → 𝐶 ∈ 𝑊)
2		imasng 6077	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐴 “ {𝐵}) = {𝑥 ∣ 𝐵𝐴𝑥})
3	2	adantr 485	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐴 “ {𝐵}) = {𝑥 ∣ 𝐵𝐴𝑥})
4		simpr 489	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 = 𝐶) → 𝑥 = 𝐶)
5	4	breq2d 5117	. 2 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ 𝑥 = 𝐶) → (𝐵𝐴𝑥 ↔ 𝐵𝐴𝐶))
6	1, 3, 5	elabd2 3632	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  {cab 2743  {csn 4585   class class class wbr 5105   “ cima 5655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-cnv 5660  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665

This theorem is referenced by:  elimasn1  6081  elimasng  6082  elimasni  6084  elinisegg  6086