Theorem inopab 5695

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 5690	. . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		relin1 5679	. . 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 5690	. 2 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}
5		sban 2082	. . . 4 ⊢ ([𝑤 / 𝑦]([𝑧 / 𝑥]𝜑 ∧ [𝑧 / 𝑥]𝜓) ↔ ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜓))
6		sban 2082	. . . . 5 ⊢ ([𝑧 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑧 / 𝑥]𝜑 ∧ [𝑧 / 𝑥]𝜓))
7	6	sbbii 2077	. . . 4 ⊢ ([𝑤 / 𝑦][𝑧 / 𝑥](𝜑 ∧ 𝜓) ↔ [𝑤 / 𝑦]([𝑧 / 𝑥]𝜑 ∧ [𝑧 / 𝑥]𝜓))
8		opelopabsbALT 5408	. . . . 5 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
9		opelopabsbALT 5408	. . . . 5 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜓)
10	8, 9	anbi12i 628	. . . 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 306	. . 3 ⊢ ((⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ [𝑤 / 𝑦][𝑧 / 𝑥](𝜑 ∧ 𝜓))
12		elin 4168	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
13		opelopabsbALT 5408	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)} ↔ [𝑤 / 𝑦][𝑧 / 𝑥](𝜑 ∧ 𝜓))
14	11, 12, 13	3bitr4i 305	. 2 ⊢ (⟨𝑧, 𝑤⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)})
15	3, 4, 14	eqrelriiv 5657	1 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}

Syntax hints:   ∧ wa 398   = wceq 1533  [wsb 2065   ∈ wcel 2110   ∩ cin 3934  ⟨cop 4566  {copab 5120  Rel wrel 5554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-opab 5121  df-xp 5555  df-rel 5556

This theorem is referenced by:  inxp  5697  resopab  5896  fndmin  6809  cnvoprab  7752  epinid0  9058  cnvepnep  9065  wemapwe  9154  dfiso2  17036  frgpuplem  18892  ltbwe  20247  opsrtoslem1  20258  pjfval2  20847  lgsquadlem3  25952  br1cosscnvxrn  35708  1cosscnvxrn  35709  dnwech  39641  fgraphopab  39803