Theorem inopab 2495

Description: Intersection of two ordered pair class abstractions.

Assertion

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

Distinct variable group(s):   x,y

Proof of Theorem inopab

Step	Hyp	Ref	Expression
1		relopab 2494	. . 3 ⊢ Rel {⟨x, y⟩∣φ}
2		relin 2491	. . 3 ⊢ (Rel {⟨x, y⟩∣φ} → Rel ({⟨x, y⟩∣φ} ∩ {⟨x, y⟩∣ψ}))
3	1, 2	ax-mp 6	. 2 ⊢ Rel ({⟨x, y⟩∣φ} ∩ {⟨x, y⟩∣ψ})
4		relopab 2494	. 2 ⊢ Rel {⟨x, y⟩∣(φ ∧ ψ)}
5		sban 889	. . . 4 ⊢ ([w / y]([z / x]φ ∧ [z / x]ψ) ↔ ([w / y][z / x]φ ∧ [w / y][z / x]ψ))
6		sban 889	. . . . 5 ⊢ ([z / x](φ ∧ ψ) ↔ ([z / x]φ ∧ [z / x]ψ))
7	6	bisb 855	. . . 4 ⊢ ([w / y][z / x](φ ∧ ψ) ↔ [w / y]([z / x]φ ∧ [z / x]ψ))
8		opabsb 2114	. . . . 5 ⊢ (⟨z, w⟩ ∈ {⟨x, y⟩∣φ} ↔ [w / y][z / x]φ)
9		opabsb 2114	. . . . 5 ⊢ (⟨z, w⟩ ∈ {⟨x, y⟩∣ψ} ↔ [w / y][z / x]ψ)
10	8, 9	anbi12i 369	. . . 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	3bitr4r 159	. . 3 ⊢ ((⟨z, w⟩ ∈ {⟨x, y⟩∣φ} ∧ ⟨z, w⟩ ∈ {⟨x, y⟩∣ψ}) ↔ [w / y][z / x](φ ∧ ψ))
12		elin 1635	. . 3 ⊢ (⟨z, w⟩ ∈ ({⟨x, y⟩∣φ} ∩ {⟨x, y⟩∣ψ}) ↔ (⟨z, w⟩ ∈ {⟨x, y⟩∣φ} ∧ ⟨z, w⟩ ∈ {⟨x, y⟩∣ψ}))
13		opabsb 2114	. . 3 ⊢ (⟨z, w⟩ ∈ {⟨x, y⟩∣(φ ∧ ψ)} ↔ [w / y][z / x](φ ∧ ψ))
14	11, 12, 13	3bitr4 158	. 2 ⊢ (⟨z, w⟩ ∈ ({⟨x, y⟩∣φ} ∩ {⟨x, y⟩∣ψ}) ↔ ⟨z, w⟩ ∈ {⟨x, y⟩∣(φ ∧ ψ)})
15	3, 4, 14	cleqreli 2484	1 ⊢ ({⟨x, y⟩∣φ} ∩ {⟨x, y⟩∣ψ}) = {⟨x, y⟩∣(φ ∧ ψ)}

Syntax hints:   ∧ wa 196  [wsb 852   = wceq 1091   ∈ wcel 1092   ∩ cin 1486  ⟨cop 1810  {copab 2055  Rel wrel 2415

This theorem is referenced by:  resopab 2598

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424  df-rel 2425

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