Theorem inopab 4851

Description: Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)

Assertion

Ref	Expression
inopab	⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem inopab

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopab 4845	. . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		relin1 4834	. . 3 ⊢ (Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} → Rel ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
3	1, 2	ax-mp 5	. 2 ⊢ Rel ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
4		relopab 4845	. 2 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}
5		sban 2006	. . . 4 ⊢ ([𝑤 / 𝑦]([𝑧 / 𝑥]𝜑 ∧ [𝑧 / 𝑥]𝜓) ↔ ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜓))
6		sban 2006	. . . . 5 ⊢ ([𝑧 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑧 / 𝑥]𝜑 ∧ [𝑧 / 𝑥]𝜓))
7	6	sbbii 1811	. . . 4 ⊢ ([𝑤 / 𝑦][𝑧 / 𝑥](𝜑 ∧ 𝜓) ↔ [𝑤 / 𝑦]([𝑧 / 𝑥]𝜑 ∧ [𝑧 / 𝑥]𝜓))
8		opelopabsbALT 4346	. . . . 5 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
9		opelopabsbALT 4346	. . . . 5 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜓)
10	8, 9	anbi12i 460	. . . 4 ⊢ ((⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜓))
11	5, 7, 10	3bitr4ri 213	. . 3 ⊢ ((⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ [𝑤 / 𝑦][𝑧 / 𝑥](𝜑 ∧ 𝜓))
12		elin 3387	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
13		opelopabsbALT 4346	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)} ↔ [𝑤 / 𝑦][𝑧 / 𝑥](𝜑 ∧ 𝜓))
14	11, 12, 13	3bitr4i 212	. 2 ⊢ (⟨𝑧, 𝑤⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)})
15	3, 4, 14	eqrelriiv 4810	1 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}

Syntax hints:   ∧ wa 104   = wceq 1395  [wsb 1808   ∈ wcel 2200   ∩ cin 3196  ⟨cop 3669  {copab 4143  Rel wrel 4721

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4145  df-xp 4722  df-rel 4723

This theorem is referenced by:  inxp  4853  resopab  5045  cnvin  5132  fndmin  5735  enq0enq  7606  lgsquadlem3  15743