Theorem elsnres 4996

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 4995	. 2 ⊢ (𝐴 ∈ (𝐵 ↾ {𝐶}) ↔ ∃𝑥 ∈ {𝐶}∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
2		rexcom4 2795	. 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 3819	. . . . . 6 ⊢ (𝑥 = 𝐶 → ⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝑦⟩)
5	4	eqeq2d 2217	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝐶, 𝑦⟩))
6	4	eleq1d 2274	. . . . 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 3677	. . 3 ⊢ (∃𝑥 ∈ {𝐶} (𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (𝐴 = ⟨𝐶, 𝑦⟩ ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐵))
9	8	exbii 1628	. 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 1373  ∃wex 1515   ∈ wcel 2176  ∃wrex 2485  Vcvv 2772  {csn 3633  ⟨cop 3636   ↾ cres 4677

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-opab 4106  df-xp 4681  df-rel 4682  df-res 4687

This theorem is referenced by: (None)