Theorem elsnres 4997

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 4996	. 2 ⊢ (A ∈ (B ↾ {C}) ↔ ∃x ∈ {C}∃y(A = ⟨x, y⟩ ∧ ⟨x, y⟩ ∈ B))
2		rexcom4 2879	. 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 4579	. . . . . 6 ⊢ (x = C → ⟨x, y⟩ = ⟨C, y⟩)
5	4	eqeq2d 2364	. . . . 5 ⊢ (x = C → (A = ⟨x, y⟩ ↔ A = ⟨C, y⟩))
6	4	eleq1d 2419	. . . . 5 ⊢ (x = C → (⟨x, y⟩ ∈ B ↔ ⟨C, y⟩ ∈ B))
7	5, 6	anbi12d 691	. . . 4 ⊢ (x = C → ((A = ⟨x, y⟩ ∧ ⟨x, y⟩ ∈ B) ↔ (A = ⟨C, y⟩ ∧ ⟨C, y⟩ ∈ B)))
8	3, 7	rexsn 3769	. . 3 ⊢ (∃x ∈ {C} (A = ⟨x, y⟩ ∧ ⟨x, y⟩ ∈ B) ↔ (A = ⟨C, y⟩ ∧ ⟨C, y⟩ ∈ B))
9	8	exbii 1582	. 2 ⊢ (∃y∃x ∈ {C} (A = ⟨x, y⟩ ∧ ⟨x, y⟩ ∈ B) ↔ ∃y(A = ⟨C, y⟩ ∧ ⟨C, y⟩ ∈ B))
10	1, 2, 9	3bitri 262	1 ⊢ (A ∈ (B ↾ {C}) ↔ ∃y(A = ⟨C, y⟩ ∧ ⟨C, y⟩ ∈ B))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860  {csn 3738  ⟨cop 4562   ↾ cres 4775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-xp 4785  df-res 4789

This theorem is referenced by: (None)