Theorem eldmressn 47595

Description: Element of the domain of a restriction to a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)

Assertion

Ref	Expression
eldmressn	⊢ (𝐵 ∈ dom (𝐹 ↾ {𝐴}) → 𝐵 = 𝐴)

Proof of Theorem eldmressn

Step	Hyp	Ref	Expression
1		elin 3920	. . 3 ⊢ (𝐵 ∈ ({𝐴} ∩ dom 𝐹) ↔ (𝐵 ∈ {𝐴} ∧ 𝐵 ∈ dom 𝐹))
2		elsni 4598	. . . 4 ⊢ (𝐵 ∈ {𝐴} → 𝐵 = 𝐴)
3	2	adantr 484	. . 3 ⊢ ((𝐵 ∈ {𝐴} ∧ 𝐵 ∈ dom 𝐹) → 𝐵 = 𝐴)
4	1, 3	sylbi 219	. 2 ⊢ (𝐵 ∈ ({𝐴} ∩ dom 𝐹) → 𝐵 = 𝐴)
5		dmres 5996	. 2 ⊢ dom (𝐹 ↾ {𝐴}) = ({𝐴} ∩ dom 𝐹)
6	4, 5	eleq2s 2879	1 ⊢ (𝐵 ∈ dom (𝐹 ↾ {𝐴}) → 𝐵 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ∩ cin 3903  {csn 4581  dom cdm 5645   ↾ cres 5647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-dm 5655  df-res 5657

This theorem is referenced by:  dfdfat2  47686