Theorem elres 4996

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

Assertion

Ref	Expression
elres	⊢ (A ∈ (B ↾ C) ↔ ∃x ∈ C ∃y(A = ⟨x, y⟩ ∧ ⟨x, y⟩ ∈ B))

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

Proof of Theorem elres

Step	Hyp	Ref	Expression
1		eleq1 2413	. . . . . 6 ⊢ (A = ⟨x, y⟩ → (A ∈ (B ↾ C) ↔ ⟨x, y⟩ ∈ (B ↾ C)))
2		opelres 4951	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ (B ↾ C) ↔ (⟨x, y⟩ ∈ B ∧ x ∈ C))
3		ancom 437	. . . . . . 7 ⊢ ((⟨x, y⟩ ∈ B ∧ x ∈ C) ↔ (x ∈ C ∧ ⟨x, y⟩ ∈ B))
4	2, 3	bitri 240	. . . . . 6 ⊢ (⟨x, y⟩ ∈ (B ↾ C) ↔ (x ∈ C ∧ ⟨x, y⟩ ∈ B))
5	1, 4	syl6bb 252	. . . . 5 ⊢ (A = ⟨x, y⟩ → (A ∈ (B ↾ C) ↔ (x ∈ C ∧ ⟨x, y⟩ ∈ B)))
6	5	pm5.32i 618	. . . 4 ⊢ ((A = ⟨x, y⟩ ∧ A ∈ (B ↾ C)) ↔ (A = ⟨x, y⟩ ∧ (x ∈ C ∧ ⟨x, y⟩ ∈ B)))
7		an12 772	. . . 4 ⊢ ((A = ⟨x, y⟩ ∧ (x ∈ C ∧ ⟨x, y⟩ ∈ B)) ↔ (x ∈ C ∧ (A = ⟨x, y⟩ ∧ ⟨x, y⟩ ∈ B)))
8	6, 7	bitri 240	. . 3 ⊢ ((A = ⟨x, y⟩ ∧ A ∈ (B ↾ C)) ↔ (x ∈ C ∧ (A = ⟨x, y⟩ ∧ ⟨x, y⟩ ∈ B)))
9	8	2exbii 1583	. 2 ⊢ (∃x∃y(A = ⟨x, y⟩ ∧ A ∈ (B ↾ C)) ↔ ∃x∃y(x ∈ C ∧ (A = ⟨x, y⟩ ∧ ⟨x, y⟩ ∈ B)))
10		opeqex 4622	. . . 4 ⊢ (A ∈ (B ↾ C) → ∃x∃y A = ⟨x, y⟩)
11	10	pm4.71ri 614	. . 3 ⊢ (A ∈ (B ↾ C) ↔ (∃x∃y A = ⟨x, y⟩ ∧ A ∈ (B ↾ C)))
12		19.41vv 1902	. . 3 ⊢ (∃x∃y(A = ⟨x, y⟩ ∧ A ∈ (B ↾ C)) ↔ (∃x∃y A = ⟨x, y⟩ ∧ A ∈ (B ↾ C)))
13	11, 12	bitr4i 243	. 2 ⊢ (A ∈ (B ↾ C) ↔ ∃x∃y(A = ⟨x, y⟩ ∧ A ∈ (B ↾ C)))
14		df-rex 2621	. . 3 ⊢ (∃x ∈ C ∃y(A = ⟨x, y⟩ ∧ ⟨x, y⟩ ∈ B) ↔ ∃x(x ∈ C ∧ ∃y(A = ⟨x, y⟩ ∧ ⟨x, y⟩ ∈ B)))
15		exdistr 1906	. . 3 ⊢ (∃x∃y(x ∈ C ∧ (A = ⟨x, y⟩ ∧ ⟨x, y⟩ ∈ B)) ↔ ∃x(x ∈ C ∧ ∃y(A = ⟨x, y⟩ ∧ ⟨x, y⟩ ∈ B)))
16	14, 15	bitr4i 243	. 2 ⊢ (∃x ∈ C ∃y(A = ⟨x, y⟩ ∧ ⟨x, y⟩ ∈ B) ↔ ∃x∃y(x ∈ C ∧ (A = ⟨x, y⟩ ∧ ⟨x, y⟩ ∈ B)))
17	9, 13, 16	3bitr4i 268	1 ⊢ (A ∈ (B ↾ C) ↔ ∃x ∈ C ∃y(A = ⟨x, y⟩ ∧ ⟨x, y⟩ ∈ B))

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  ⟨cop 4562   ↾ cres 4775

This proof depends on 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 proof 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 used by:  elsnres  4997