Theorem eloprabga 5579

Description: The law of concretion for operation class abstraction. Compare elopab 4697. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 18-Jun-2012.) Removed unnecessary distinct variable requirements. (Revised by Mario Carneiro, 19-Dec-2013.)

Hypothesis

Ref	Expression
eloprabga.1	⊢ ((x = A ∧ y = B ∧ z = C) → (φ ↔ ψ))

Assertion

Ref	Expression
eloprabga	⊢ ((A ∈ V ∧ B ∈ W ∧ C ∈ X) → (⟨⟨A, B⟩, C⟩ ∈ {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ ψ))

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

Allowed substitution hints:   φ(x,y,z)   V(x,y,z)   W(x,y,z)   X(x,y,z)

Proof of Theorem eloprabga

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2868	. 2 ⊢ (A ∈ V → A ∈ V)
2		elex 2868	. 2 ⊢ (B ∈ W → B ∈ V)
3		elex 2868	. 2 ⊢ (C ∈ X → C ∈ V)
4		opexg 4588	. . . . 5 ⊢ ((A ∈ V ∧ B ∈ V) → ⟨A, B⟩ ∈ V)
5		opexg 4588	. . . . 5 ⊢ ((⟨A, B⟩ ∈ V ∧ C ∈ V) → ⟨⟨A, B⟩, C⟩ ∈ V)
6	4, 5	sylan 457	. . . 4 ⊢ (((A ∈ V ∧ B ∈ V) ∧ C ∈ V) → ⟨⟨A, B⟩, C⟩ ∈ V)
7	6	3impa 1146	. . 3 ⊢ ((A ∈ V ∧ B ∈ V ∧ C ∈ V) → ⟨⟨A, B⟩, C⟩ ∈ V)
8		eqeq1 2359	. . . . . . . . . . 11 ⊢ (w = ⟨⟨A, B⟩, C⟩ → (w = ⟨⟨x, y⟩, z⟩ ↔ ⟨⟨A, B⟩, C⟩ = ⟨⟨x, y⟩, z⟩))
9		eqcom 2355	. . . . . . . . . . . 12 ⊢ (⟨⟨A, B⟩, C⟩ = ⟨⟨x, y⟩, z⟩ ↔ ⟨⟨x, y⟩, z⟩ = ⟨⟨A, B⟩, C⟩)
10		opth 4603	. . . . . . . . . . . . . 14 ⊢ (⟨x, y⟩ = ⟨A, B⟩ ↔ (x = A ∧ y = B))
11	10	anbi1i 676	. . . . . . . . . . . . 13 ⊢ ((⟨x, y⟩ = ⟨A, B⟩ ∧ z = C) ↔ ((x = A ∧ y = B) ∧ z = C))
12		opth 4603	. . . . . . . . . . . . 13 ⊢ (⟨⟨x, y⟩, z⟩ = ⟨⟨A, B⟩, C⟩ ↔ (⟨x, y⟩ = ⟨A, B⟩ ∧ z = C))
13		df-3an 936	. . . . . . . . . . . . 13 ⊢ ((x = A ∧ y = B ∧ z = C) ↔ ((x = A ∧ y = B) ∧ z = C))
14	11, 12, 13	3bitr4i 268	. . . . . . . . . . . 12 ⊢ (⟨⟨x, y⟩, z⟩ = ⟨⟨A, B⟩, C⟩ ↔ (x = A ∧ y = B ∧ z = C))
15	9, 14	bitri 240	. . . . . . . . . . 11 ⊢ (⟨⟨A, B⟩, C⟩ = ⟨⟨x, y⟩, z⟩ ↔ (x = A ∧ y = B ∧ z = C))
16	8, 15	syl6bb 252	. . . . . . . . . 10 ⊢ (w = ⟨⟨A, B⟩, C⟩ → (w = ⟨⟨x, y⟩, z⟩ ↔ (x = A ∧ y = B ∧ z = C)))
17	16	anbi1d 685	. . . . . . . . 9 ⊢ (w = ⟨⟨A, B⟩, C⟩ → ((w = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ ((x = A ∧ y = B ∧ z = C) ∧ φ)))
18		eloprabga.1	. . . . . . . . . 10 ⊢ ((x = A ∧ y = B ∧ z = C) → (φ ↔ ψ))
19	18	pm5.32i 618	. . . . . . . . 9 ⊢ (((x = A ∧ y = B ∧ z = C) ∧ φ) ↔ ((x = A ∧ y = B ∧ z = C) ∧ ψ))
20	17, 19	syl6bb 252	. . . . . . . 8 ⊢ (w = ⟨⟨A, B⟩, C⟩ → ((w = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ ((x = A ∧ y = B ∧ z = C) ∧ ψ)))
21	20	3exbidv 1629	. . . . . . 7 ⊢ (w = ⟨⟨A, B⟩, C⟩ → (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ ∃x∃y∃z((x = A ∧ y = B ∧ z = C) ∧ ψ)))
22	21	adantl 452	. . . . . 6 ⊢ (((A ∈ V ∧ B ∈ V ∧ C ∈ V) ∧ w = ⟨⟨A, B⟩, C⟩) → (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ ∃x∃y∃z((x = A ∧ y = B ∧ z = C) ∧ ψ)))
23		df-oprab 5529	. . . . . . . . . 10 ⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} = {w ∣ ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ)}
24	23	eleq2i 2417	. . . . . . . . 9 ⊢ (w ∈ {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ w ∈ {w ∣ ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ)})
25		abid 2341	. . . . . . . . 9 ⊢ (w ∈ {w ∣ ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ)} ↔ ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ))
26	24, 25	bitr2i 241	. . . . . . . 8 ⊢ (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ w ∈ {⟨⟨x, y⟩, z⟩ ∣ φ})
27		eleq1 2413	. . . . . . . 8 ⊢ (w = ⟨⟨A, B⟩, C⟩ → (w ∈ {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ ⟨⟨A, B⟩, C⟩ ∈ {⟨⟨x, y⟩, z⟩ ∣ φ}))
28	26, 27	syl5bb 248	. . . . . . 7 ⊢ (w = ⟨⟨A, B⟩, C⟩ → (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ ⟨⟨A, B⟩, C⟩ ∈ {⟨⟨x, y⟩, z⟩ ∣ φ}))
29	28	adantl 452	. . . . . 6 ⊢ (((A ∈ V ∧ B ∈ V ∧ C ∈ V) ∧ w = ⟨⟨A, B⟩, C⟩) → (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ ⟨⟨A, B⟩, C⟩ ∈ {⟨⟨x, y⟩, z⟩ ∣ φ}))
30		isset 2864	. . . . . . . . . . . 12 ⊢ (A ∈ V ↔ ∃x x = A)
31		isset 2864	. . . . . . . . . . . 12 ⊢ (B ∈ V ↔ ∃y y = B)
32		isset 2864	. . . . . . . . . . . 12 ⊢ (C ∈ V ↔ ∃z z = C)
33	30, 31, 32	3anbi123i 1140	. . . . . . . . . . 11 ⊢ ((A ∈ V ∧ B ∈ V ∧ C ∈ V) ↔ (∃x x = A ∧ ∃y y = B ∧ ∃z z = C))
34		eeeanv 1914	. . . . . . . . . . 11 ⊢ (∃x∃y∃z(x = A ∧ y = B ∧ z = C) ↔ (∃x x = A ∧ ∃y y = B ∧ ∃z z = C))
35	33, 34	bitr4i 243	. . . . . . . . . 10 ⊢ ((A ∈ V ∧ B ∈ V ∧ C ∈ V) ↔ ∃x∃y∃z(x = A ∧ y = B ∧ z = C))
36	35	biimpi 186	. . . . . . . . 9 ⊢ ((A ∈ V ∧ B ∈ V ∧ C ∈ V) → ∃x∃y∃z(x = A ∧ y = B ∧ z = C))
37	36	biantrurd 494	. . . . . . . 8 ⊢ ((A ∈ V ∧ B ∈ V ∧ C ∈ V) → (ψ ↔ (∃x∃y∃z(x = A ∧ y = B ∧ z = C) ∧ ψ)))
38		19.41vvv 1903	. . . . . . . 8 ⊢ (∃x∃y∃z((x = A ∧ y = B ∧ z = C) ∧ ψ) ↔ (∃x∃y∃z(x = A ∧ y = B ∧ z = C) ∧ ψ))
39	37, 38	syl6rbbr 255	. . . . . . 7 ⊢ ((A ∈ V ∧ B ∈ V ∧ C ∈ V) → (∃x∃y∃z((x = A ∧ y = B ∧ z = C) ∧ ψ) ↔ ψ))
40	39	adantr 451	. . . . . 6 ⊢ (((A ∈ V ∧ B ∈ V ∧ C ∈ V) ∧ w = ⟨⟨A, B⟩, C⟩) → (∃x∃y∃z((x = A ∧ y = B ∧ z = C) ∧ ψ) ↔ ψ))
41	22, 29, 40	3bitr3d 274	. . . . 5 ⊢ (((A ∈ V ∧ B ∈ V ∧ C ∈ V) ∧ w = ⟨⟨A, B⟩, C⟩) → (⟨⟨A, B⟩, C⟩ ∈ {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ ψ))
42	41	expcom 424	. . . 4 ⊢ (w = ⟨⟨A, B⟩, C⟩ → ((A ∈ V ∧ B ∈ V ∧ C ∈ V) → (⟨⟨A, B⟩, C⟩ ∈ {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ ψ)))
43	42	vtocleg 2926	. . 3 ⊢ (⟨⟨A, B⟩, C⟩ ∈ V → ((A ∈ V ∧ B ∈ V ∧ C ∈ V) → (⟨⟨A, B⟩, C⟩ ∈ {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ ψ)))
44	7, 43	mpcom 32	. 2 ⊢ ((A ∈ V ∧ B ∈ V ∧ C ∈ V) → (⟨⟨A, B⟩, C⟩ ∈ {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ ψ))
45	1, 2, 3, 44	syl3an 1224	1 ⊢ ((A ∈ V ∧ B ∈ W ∧ C ∈ X) → (⟨⟨A, B⟩, C⟩ ∈ {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ ψ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  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:  eloprabg  5580  ovigg  5597