Theorem dfoprab2 5559

Description: Class abstraction for operations in terms of class abstraction of ordered pairs. (Contributed by set.mm contributors, 12-Mar-1995.)

Assertion

Ref	Expression
dfoprab2	⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)}

Distinct variable groups:   x,z,w   y,z,w   φ,w

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

Proof of Theorem dfoprab2

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		excom 1741	. . . 4 ⊢ (∃z∃w∃x∃y(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ ∃w∃z∃x∃y(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)))
2		exrot4 1745	. . . . 5 ⊢ (∃z∃w∃x∃y(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ ∃x∃y∃z∃w(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)))
3		an12 772	. . . . . . . 8 ⊢ ((v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ (w = ⟨x, y⟩ ∧ (v = ⟨w, z⟩ ∧ φ)))
4	3	exbii 1582	. . . . . . 7 ⊢ (∃w(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ ∃w(w = ⟨x, y⟩ ∧ (v = ⟨w, z⟩ ∧ φ)))
5		vex 2863	. . . . . . . . 9 ⊢ x ∈ V
6		vex 2863	. . . . . . . . 9 ⊢ y ∈ V
7	5, 6	opex 4589	. . . . . . . 8 ⊢ ⟨x, y⟩ ∈ V
8		opeq1 4579	. . . . . . . . . 10 ⊢ (w = ⟨x, y⟩ → ⟨w, z⟩ = ⟨⟨x, y⟩, z⟩)
9	8	eqeq2d 2364	. . . . . . . . 9 ⊢ (w = ⟨x, y⟩ → (v = ⟨w, z⟩ ↔ v = ⟨⟨x, y⟩, z⟩))
10	9	anbi1d 685	. . . . . . . 8 ⊢ (w = ⟨x, y⟩ → ((v = ⟨w, z⟩ ∧ φ) ↔ (v = ⟨⟨x, y⟩, z⟩ ∧ φ)))
11	7, 10	ceqsexv 2895	. . . . . . 7 ⊢ (∃w(w = ⟨x, y⟩ ∧ (v = ⟨w, z⟩ ∧ φ)) ↔ (v = ⟨⟨x, y⟩, z⟩ ∧ φ))
12	4, 11	bitri 240	. . . . . 6 ⊢ (∃w(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ (v = ⟨⟨x, y⟩, z⟩ ∧ φ))
13	12	3exbii 1584	. . . . 5 ⊢ (∃x∃y∃z∃w(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ ∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ))
14	2, 13	bitri 240	. . . 4 ⊢ (∃z∃w∃x∃y(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ ∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ))
15		19.42vv 1907	. . . . 5 ⊢ (∃x∃y(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ (v = ⟨w, z⟩ ∧ ∃x∃y(w = ⟨x, y⟩ ∧ φ)))
16	15	2exbii 1583	. . . 4 ⊢ (∃w∃z∃x∃y(v = ⟨w, z⟩ ∧ (w = ⟨x, y⟩ ∧ φ)) ↔ ∃w∃z(v = ⟨w, z⟩ ∧ ∃x∃y(w = ⟨x, y⟩ ∧ φ)))
17	1, 14, 16	3bitr3i 266	. . 3 ⊢ (∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ ∃w∃z(v = ⟨w, z⟩ ∧ ∃x∃y(w = ⟨x, y⟩ ∧ φ)))
18	17	abbii 2466	. 2 ⊢ {v ∣ ∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)} = {v ∣ ∃w∃z(v = ⟨w, z⟩ ∧ ∃x∃y(w = ⟨x, y⟩ ∧ φ))}
19		df-oprab 5529	. 2 ⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} = {v ∣ ∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)}
20		df-opab 4624	. 2 ⊢ {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)} = {v ∣ ∃w∃z(v = ⟨w, z⟩ ∧ ∃x∃y(w = ⟨x, y⟩ ∧ φ))}
21	18, 19, 20	3eqtr4i 2383	1 ⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)}

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642  {cab 2339  ⟨cop 4562  {copab 4623  {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-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-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  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-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-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-opab 4624  df-oprab 5529

This theorem is referenced by:  cbvoprab1  5568  cbvoprab12  5570  cbvoprab3  5572  dmoprab  5575  rnoprab  5577  ssoprab2i  5581  resoprab  5582  funoprabg  5584  fnov  5592  ov6g  5601  mpt2mptx  5709  dfswap3  5729