Theorem oprabid2 5647

Description: Identity law for operator abstractions. (Contributed by Scott Fenton, 19-Apr-2021.)

Assertion

Ref	Expression
oprabid2	⊢ {⟨⟨x, y⟩, z⟩ ∣ ⟨⟨x, y⟩, z⟩ ∈ A} = A

Distinct variable group:   x,y,z,A

Proof of Theorem oprabid2

Dummy variables w t u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2863	. . 3 ⊢ w ∈ V
2		vex 2863	. . 3 ⊢ t ∈ V
3		vex 2863	. . 3 ⊢ u ∈ V
4		opeq1 4579	. . . . . 6 ⊢ (x = w → ⟨x, y⟩ = ⟨w, y⟩)
5	4	opeq1d 4585	. . . . 5 ⊢ (x = w → ⟨⟨x, y⟩, z⟩ = ⟨⟨w, y⟩, z⟩)
6	5	eleq1d 2419	. . . 4 ⊢ (x = w → (⟨⟨x, y⟩, z⟩ ∈ A ↔ ⟨⟨w, y⟩, z⟩ ∈ A))
7		opeq2 4580	. . . . . 6 ⊢ (y = t → ⟨w, y⟩ = ⟨w, t⟩)
8	7	opeq1d 4585	. . . . 5 ⊢ (y = t → ⟨⟨w, y⟩, z⟩ = ⟨⟨w, t⟩, z⟩)
9	8	eleq1d 2419	. . . 4 ⊢ (y = t → (⟨⟨w, y⟩, z⟩ ∈ A ↔ ⟨⟨w, t⟩, z⟩ ∈ A))
10		opeq2 4580	. . . . 5 ⊢ (z = u → ⟨⟨w, t⟩, z⟩ = ⟨⟨w, t⟩, u⟩)
11	10	eleq1d 2419	. . . 4 ⊢ (z = u → (⟨⟨w, t⟩, z⟩ ∈ A ↔ ⟨⟨w, t⟩, u⟩ ∈ A))
12	6, 9, 11	eloprabg 5580	. . 3 ⊢ ((w ∈ V ∧ t ∈ V ∧ u ∈ V) → (⟨⟨w, t⟩, u⟩ ∈ {⟨⟨x, y⟩, z⟩ ∣ ⟨⟨x, y⟩, z⟩ ∈ A} ↔ ⟨⟨w, t⟩, u⟩ ∈ A))
13	1, 2, 3, 12	mp3an 1277	. 2 ⊢ (⟨⟨w, t⟩, u⟩ ∈ {⟨⟨x, y⟩, z⟩ ∣ ⟨⟨x, y⟩, z⟩ ∈ A} ↔ ⟨⟨w, t⟩, u⟩ ∈ A)
14	13	eqoprriv 4854	1 ⊢ {⟨⟨x, y⟩, z⟩ ∣ ⟨⟨x, y⟩, z⟩ ∈ A} = A

Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  Vcvv 2860  ⟨cop 4562  {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-oprab 5529

This theorem is referenced by:  oprabbi2i  5648