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 2878	. 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 4578	. . . . . 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 3768	. . 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 2615  Vcvv 2859  {csn 3737  ⟨cop 4561   ↾ cres 4774

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-xp 4784  df-res 4788

This theorem is referenced by: (None)