Theorem inopab 5252

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 5247	. . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		relin1 5236	. . 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 5247	. 2 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}
5		sban 2399	. . . 4 ⊢ ([𝑤 / 𝑦]([𝑧 / 𝑥]𝜑 ∧ [𝑧 / 𝑥]𝜓) ↔ ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜓))
6		sban 2399	. . . . 5 ⊢ ([𝑧 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑧 / 𝑥]𝜑 ∧ [𝑧 / 𝑥]𝜓))
7	6	sbbii 1887	. . . 4 ⊢ ([𝑤 / 𝑦][𝑧 / 𝑥](𝜑 ∧ 𝜓) ↔ [𝑤 / 𝑦]([𝑧 / 𝑥]𝜑 ∧ [𝑧 / 𝑥]𝜓))
8		opelopabsbALT 4984	. . . . 5 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
9		opelopabsbALT 4984	. . . . 5 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜓)
10	8, 9	anbi12i 733	. . . 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 293	. . 3 ⊢ ((⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ [𝑤 / 𝑦][𝑧 / 𝑥](𝜑 ∧ 𝜓))
12		elin 3796	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
13		opelopabsbALT 4984	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)} ↔ [𝑤 / 𝑦][𝑧 / 𝑥](𝜑 ∧ 𝜓))
14	11, 12, 13	3bitr4i 292	. 2 ⊢ (⟨𝑧, 𝑤⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)})
15	3, 4, 14	eqrelriiv 5214	1 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}

Syntax hints:   ∧ wa 384   = wceq 1483  [wsb 1880   ∈ wcel 1990   ∩ cin 3573  ⟨cop 4183  {copab 4712  Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  inxp  5254  resopab  5446  fndmin  6324  wemapwe  8594  dfiso2  16432  frgpuplem  18185  ltbwe  19472  opsrtoslem1  19484  pjfval2  20053  lgsquadlem3  25107  dnwech  37618  fgraphopab  37788