Theorem elsnres 4946

Description: Memebership in restriction to a singleton. (Contributed by Scott Fenton, 17-Mar-2011.)

Hypothesis

Ref	Expression
elsnres.1	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
elsnres	⊢ (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑦(𝐴 = ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐵))

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝐶

Proof of Theorem elsnres

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elres 4945	. 2 ⊢ (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑥 ∈ {𝐶}∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
2		rexcom4 2762	. 2 ⊢ (∃𝑥 ∈ {𝐶}∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦∃𝑥 ∈ {𝐶} (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
3		elsnres.1	. . . 4 ⊢ 𝐶 ∈ V
4		opeq1 3780	. . . . . 6 ⊢ (𝑥 = 𝐶 → ⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝑦⟩)
5	4	eqeq2d 2189	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝐶, 𝑦⟩))
6	4	eleq1d 2246	. . . . 5 ⊢ (𝑥 = 𝐶 → (⟨𝑥, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝐶, 𝑦⟩ ∈ 𝐵))
7	5, 6	anbi12d 473	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (𝐴 = ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐵)))
8	3, 7	rexsn 3638	. . 3 ⊢ (∃𝑥 ∈ {𝐶} (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (𝐴 = ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐵))
9	8	exbii 1605	. 2 ⊢ (∃𝑦∃𝑥 ∈ {𝐶} (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑦(𝐴 = ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐵))
10	1, 2, 9	3bitri 206	1 ⊢ (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑦(𝐴 = ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐵))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∃wrex 2456  Vcvv 2739  {csn 3594  ⟨cop 3597   ↾ cres 4630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-opab 4067  df-xp 4634  df-rel 4635  df-res 4640

This theorem is referenced by: (None)