Theorem resopab2 5001

Description: Restriction of a class abstraction of ordered pairs. (Contributed by set.mm contributors, 24-Aug-2007.)

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

Allowed substitution hints:   φ(x,y)

Proof of Theorem resopab2

Step	Hyp	Ref	Expression
1		resopab 4999	. 2 ⊢ ({⟨x, y⟩ ∣ (x ∈ B ∧ φ)} ↾ A) = {⟨x, y⟩ ∣ (x ∈ A ∧ (x ∈ B ∧ φ))}
2		ssel 3267	. . . . . 6 ⊢ (A ⊆ B → (x ∈ A → x ∈ B))
3		pm4.71 611	. . . . . 6 ⊢ ((x ∈ A → x ∈ B) ↔ (x ∈ A ↔ (x ∈ A ∧ x ∈ B)))
4	2, 3	sylib 188	. . . . 5 ⊢ (A ⊆ B → (x ∈ A ↔ (x ∈ A ∧ x ∈ B)))
5	4	anbi1d 685	. . . 4 ⊢ (A ⊆ B → ((x ∈ A ∧ φ) ↔ ((x ∈ A ∧ x ∈ B) ∧ φ)))
6		anass 630	. . . 4 ⊢ (((x ∈ A ∧ x ∈ B) ∧ φ) ↔ (x ∈ A ∧ (x ∈ B ∧ φ)))
7	5, 6	syl6rbb 253	. . 3 ⊢ (A ⊆ B → ((x ∈ A ∧ (x ∈ B ∧ φ)) ↔ (x ∈ A ∧ φ)))
8	7	opabbidv 4625	. 2 ⊢ (A ⊆ B → {⟨x, y⟩ ∣ (x ∈ A ∧ (x ∈ B ∧ φ))} = {⟨x, y⟩ ∣ (x ∈ A ∧ φ)})
9	1, 8	syl5eq 2397	1 ⊢ (A ⊆ B → ({⟨x, y⟩ ∣ (x ∈ B ∧ φ)} ↾ A) = {⟨x, y⟩ ∣ (x ∈ A ∧ φ)})

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ⊆ wss 3257  {copab 4622   ↾ cres 4774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-xp 4784  df-res 4788

This theorem is referenced by:  resmpt  5696