Theorem resoprab 5582

Description: Restriction of an operation class abstraction. (Contributed by set.mm contributors, 10-Feb-2007.)

Assertion

Ref	Expression
resoprab	⊢ ({⟨⟨x, y⟩, z⟩ ∣ φ} ↾ (A × B)) = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ φ)}

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

Allowed substitution hints:   φ(x,y,z)

Proof of Theorem resoprab

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		resopab 5000	. . 3 ⊢ ({⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)} ↾ (A × B)) = {⟨w, z⟩ ∣ (w ∈ (A × B) ∧ ∃x∃y(w = ⟨x, y⟩ ∧ φ))}
2		19.42vv 1907	. . . . 5 ⊢ (∃x∃y(w ∈ (A × B) ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ (w ∈ (A × B) ∧ ∃x∃y(w = ⟨x, y⟩ ∧ φ)))
3		an12 772	. . . . . . 7 ⊢ ((w ∈ (A × B) ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ (w = ⟨x, y⟩ ∧ (w ∈ (A × B) ∧ φ)))
4		eleq1 2413	. . . . . . . . . 10 ⊢ (w = ⟨x, y⟩ → (w ∈ (A × B) ↔ ⟨x, y⟩ ∈ (A × B)))
5		opelxp 4812	. . . . . . . . . 10 ⊢ (⟨x, y⟩ ∈ (A × B) ↔ (x ∈ A ∧ y ∈ B))
6	4, 5	syl6bb 252	. . . . . . . . 9 ⊢ (w = ⟨x, y⟩ → (w ∈ (A × B) ↔ (x ∈ A ∧ y ∈ B)))
7	6	anbi1d 685	. . . . . . . 8 ⊢ (w = ⟨x, y⟩ → ((w ∈ (A × B) ∧ φ) ↔ ((x ∈ A ∧ y ∈ B) ∧ φ)))
8	7	pm5.32i 618	. . . . . . 7 ⊢ ((w = ⟨x, y⟩ ∧ (w ∈ (A × B) ∧ φ)) ↔ (w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ φ)))
9	3, 8	bitri 240	. . . . . 6 ⊢ ((w ∈ (A × B) ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ (w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ φ)))
10	9	2exbii 1583	. . . . 5 ⊢ (∃x∃y(w ∈ (A × B) ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ ∃x∃y(w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ φ)))
11	2, 10	bitr3i 242	. . . 4 ⊢ ((w ∈ (A × B) ∧ ∃x∃y(w = ⟨x, y⟩ ∧ φ)) ↔ ∃x∃y(w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ φ)))
12	11	opabbii 4627	. . 3 ⊢ {⟨w, z⟩ ∣ (w ∈ (A × B) ∧ ∃x∃y(w = ⟨x, y⟩ ∧ φ))} = {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ φ))}
13	1, 12	eqtri 2373	. 2 ⊢ ({⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)} ↾ (A × B)) = {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ φ))}
14		dfoprab2 5559	. . 3 ⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)}
15	14	reseq1i 4931	. 2 ⊢ ({⟨⟨x, y⟩, z⟩ ∣ φ} ↾ (A × B)) = ({⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)} ↾ (A × B))
16		dfoprab2 5559	. 2 ⊢ {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ φ)} = {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ ((x ∈ A ∧ y ∈ B) ∧ φ))}
17	13, 15, 16	3eqtr4i 2383	1 ⊢ ({⟨⟨x, y⟩, z⟩ ∣ φ} ↾ (A × B)) = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ φ)}

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ⟨cop 4562  {copab 4623   × cxp 4771   ↾ cres 4775  {coprab 5528

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-xp 4785  df-res 4789  df-oprab 5529

This theorem is referenced by:  resoprab2  5583