Theorem resopab 4991

Description: Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.)

Assertion

Ref	Expression
resopab	⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem resopab

Step	Hyp	Ref	Expression
1		df-res 4676	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V))
2		df-xp 4670	. . . . . 6 ⊢ (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)}
3		vex 2766	. . . . . . . 8 ⊢ 𝑦 ∈ V
4	3	biantru 302	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V))
5	4	opabbii 4101	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)}
6	2, 5	eqtr4i 2220	. . . . 5 ⊢ (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴}
7	6	ineq2i 3362	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V)) = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴})
8		incom 3356	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴}) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
9	7, 8	eqtri 2217	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V)) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
10		inopab 4799	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
11	9, 10	eqtri 2217	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
12	1, 11	eqtri 2217	1 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}

Syntax hints:   ∧ wa 104   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ∩ cin 3156  {copab 4094   × cxp 4662   ↾ cres 4666

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-opab 4096  df-xp 4670  df-rel 4671  df-res 4676

This theorem is referenced by:  resopab2  4994  opabresid  5000  mptpreima  5164  isarep2  5346  resoprab  6022  df1st2  6286  df2nd2  6287