Theorem resopab2 3455

Description: Restriction of a class abstraction of ordered pairs.

Assertion

Ref	Expression
resopab2	⊢ (A ⊆ B → ({⟨x, y⟩∣(x ∈ B ⋀ φ)} ↾ A) = {⟨x, y⟩∣(x ∈ A ⋀ φ)})

Distinct variable groups:   x,y,A   x,B,y

Proof of Theorem resopab2

Step	Hyp	Ref	Expression
1		ssel 2114	. . . . . 6 ⊢ (A ⊆ B → (x ∈ A → x ∈ B))
2		pm4.71 646	. . . . . 6 ⊢ ((x ∈ A → x ∈ B) ↔ (x ∈ A ↔ (x ∈ A ⋀ x ∈ B)))
3	1, 2	sylib 205	. . . . 5 ⊢ (A ⊆ B → (x ∈ A ↔ (x ∈ A ⋀ x ∈ B)))
4	3	anbi1d 628	. . . 4 ⊢ (A ⊆ B → ((x ∈ A ⋀ φ) ↔ ((x ∈ A ⋀ x ∈ B) ⋀ φ)))
5		anass 450	. . . 4 ⊢ (((x ∈ A ⋀ x ∈ B) ⋀ φ) ↔ (x ∈ A ⋀ (x ∈ B ⋀ φ)))
6	4, 5	syl6rbb 548	. . 3 ⊢ (A ⊆ B → ((x ∈ A ⋀ (x ∈ B ⋀ φ)) ↔ (x ∈ A ⋀ φ)))
7	6	opabbidv 2725	. 2 ⊢ (A ⊆ B → {⟨x, y⟩∣(x ∈ A ⋀ (x ∈ B ⋀ φ))} = {⟨x, y⟩∣(x ∈ A ⋀ φ)})
8		resopab 3452	. 2 ⊢ ({⟨x, y⟩∣(x ∈ B ⋀ φ)} ↾ A) = {⟨x, y⟩∣(x ∈ A ⋀ (x ∈ B ⋀ φ))}
9	7, 8	syl5eq 1566	1 ⊢ (A ⊆ B → ({⟨x, y⟩∣(x ∈ B ⋀ φ)} ↾ A) = {⟨x, y⟩∣(x ∈ A ⋀ φ)})

Syntax hints:   → wi 3   ↔ wb 153   ⋀ wa 230   = wceq 997   ∈ wcel 999   ⊆ wss 2098  {copab 2721   ↾ cres 3229

This theorem is referenced by:  geolim1i 7328  efseq0ex 7401  reeff1 7501  ipasslem7 8580

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-sep 2758  ax-pow 2798  ax-pr 2835

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-v 1859  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-pw 2454  df-sn 2464  df-pr 2465  df-op 2468  df-opab 2722  df-xp 3241  df-rel 3242  df-res 3247

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