Theorem inopab 5819

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		relopabv 5811	. . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		relin1 5802	. . 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		relopabv 5811	. 2 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}
5		sban 2117	. . . 4 ⊢ ([𝑧 / 𝑥]([𝑤 / 𝑦]𝜑 ∧ [𝑤 / 𝑦]𝜓) ↔ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓))
6		sban 2117	. . . . 5 ⊢ ([𝑤 / 𝑦](𝜑 ∧ 𝜓) ↔ ([𝑤 / 𝑦]𝜑 ∧ [𝑤 / 𝑦]𝜓))
7	6	sbbii 2113	. . . 4 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦](𝜑 ∧ 𝜓) ↔ [𝑧 / 𝑥]([𝑤 / 𝑦]𝜑 ∧ [𝑤 / 𝑦]𝜓))
8		vopelopabsb 5516	. . . . 5 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
9		vopelopabsb 5516	. . . . 5 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓)
10	8, 9	anbi12i 640	. . . 4 ⊢ ((⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜓))
11	5, 7, 10	3bitr4ri 307	. . 3 ⊢ ((⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ [𝑧 / 𝑥][𝑤 / 𝑦](𝜑 ∧ 𝜓))
12		elin 3924	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
13		vopelopabsb 5516	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)} ↔ [𝑧 / 𝑥][𝑤 / 𝑦](𝜑 ∧ 𝜓))
14	11, 12, 13	3bitr4i 306	. 2 ⊢ (⟨𝑧, 𝑤⟩ ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)})
15	3, 4, 14	eqrelriiv 5779	1 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  [wsb 2099   ∈ wcel 2146   ∩ cin 3907  ⟨cop 4598  {copab 5176  Rel wrel 5669

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2216  ax-ext 2738  ax-sep 5260  ax-pr 5407

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-opab 5177  df-xp 5670  df-rel 5671

This theorem is used by:  resopab  6039  fndmin  7044  cnvoprab  8059  epinid0  9569  cnvepnep  9579  wemapwe  9668  dfiso2  17839  frgpuplem  19852  pjfval2  21874  ltbwe  22210  opsrtoslem1  22221  lgsquadlem3  27561  disjecxrn  39093  br1cosscnvxrn  39245  1cosscnvxrn  39246  dfpetparts2  39653  dfpeters2  39655  dnwech  43807  fgraphopab  43962