Theorem eloprabg 3035

Description: The law of concretion for operation class abstraction. Compare elopab 2110.

Hypotheses

Ref	Expression
eloprabg.1	⊢ (x = A → (φ ↔ ψ))
eloprabg.2	⊢ (y = B → (ψ ↔ χ))
eloprabg.3	⊢ (z = C → (χ ↔ θ))

Assertion

Ref	Expression
eloprabg	⊢ ((A ∈ D ∧ B ∈ R ∧ C ∈ S) → (⟨⟨A, B⟩, C⟩ ∈ {⟨⟨x, y⟩, z⟩∣φ} ↔ θ))

Distinct variable group(s):   x,y,z,A   x,B,y,z   x,C,y,z   x,D,y,z   x,R,y,z   x,S,y,z   ψ,x   χ,x,y   θ,x,y,z

Proof of Theorem eloprabg

Step	Hyp	Ref	Expression
1		opex 1893	. 2 ⊢ ⟨⟨A, B⟩, C⟩ ∈ V
2		cleq1 1107	. . . . . . . . . 10 ⊢ (w = ⟨⟨A, B⟩, C⟩ → (w = ⟨⟨x, y⟩, z⟩ ↔ ⟨⟨A, B⟩, C⟩ = ⟨⟨x, y⟩, z⟩))
3		cleqcom 1103	. . . . . . . . . 10 ⊢ (⟨⟨A, B⟩, C⟩ = ⟨⟨x, y⟩, z⟩ ↔ ⟨⟨x, y⟩, z⟩ = ⟨⟨A, B⟩, C⟩)
4	2, 3	syl6bb 414	. . . . . . . . 9 ⊢ (w = ⟨⟨A, B⟩, C⟩ → (w = ⟨⟨x, y⟩, z⟩ ↔ ⟨⟨x, y⟩, z⟩ = ⟨⟨A, B⟩, C⟩))
5		visset 1350	. . . . . . . . . . 11 ⊢ x ∈ V
6		visset 1350	. . . . . . . . . . 11 ⊢ y ∈ V
7		visset 1350	. . . . . . . . . . 11 ⊢ z ∈ V
8	5, 6, 7	otthg 1900	. . . . . . . . . 10 ⊢ ((B ∈ R ∧ C ∈ S) → (⟨⟨x, y⟩, z⟩ = ⟨⟨A, B⟩, C⟩ ↔ (x = A ∧ y = B ∧ z = C)))
9	8	3adant1 597	. . . . . . . . 9 ⊢ ((A ∈ D ∧ B ∈ R ∧ C ∈ S) → (⟨⟨x, y⟩, z⟩ = ⟨⟨A, B⟩, C⟩ ↔ (x = A ∧ y = B ∧ z = C)))
10	4, 9	sylan9bbr 419	. . . . . . . 8 ⊢ (((<3ONT COLOR="#CC33CC">A ∈ D ∧ B ∈ R ∧ C ∈ S) ∧ w = ⟨⟨A, B⟩, C⟩) → (w = ⟨⟨x, y⟩, z⟩ ↔ (x = A ∧ y = B ∧ z = C)))
11	10	anbi1d 469	. . . . . . 7 ⊢ (((A ∈ D ∧ B ∈ R ∧ C ∈ S) ∧ w = ⟨⟨A, B⟩, C⟩) → ((w = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ ((x = A ∧ y = B ∧ z = C) ∧ φ)))
12		eloprabg.1	. . . . . . . . 9 ⊢ (x = A → (φ ↔ ψ))
13		eloprabg.2	. . . . . . . . 9 ⊢ (y = B → (ψ ↔ χ))
14		eloprabg.3	. . . . . . . . 9 ⊢ (z = C → (χ ↔ θ))
15	12, 13, 14	syl3an9b 634	. . . . . . . 8 ⊢ ((x = A ∧ y = B ∧ z = C) → (φ ↔ θ))
16	15	pm5.32i 489	. . . . . . 7 ⊢ (((x = A ∧ y = B ∧ z = C) ∧ φ) ↔ ((x = A ∧ y = B ∧ z = C) ∧ θ))
17	11, 16	syl6bb 414	. . . . . 6 ⊢ (((A ∈ D ∧ B ∈ R ∧ C ∈ S) ∧ w = ⟨⟨A, B⟩, C⟩) → ((w = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ ((x = A ∧ y = B ∧ z = C) ∧ θ)))
18	17	bi3exdv 939	. . . . 5 ⊢ (((A ∈ D ∧ B ∈ R ∧ C ∈ S) ∧ w = ⟨⟨A, B⟩, C⟩) → (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ ∃x∃y∃z((x = A ∧ y = B ∧ z = C) ∧ θ)))
19		eleq1 1149	. . . . . . 7 ⊢ (w = ⟨⟨A, B⟩, C⟩ → (w ∈ {⟨⟨x, y⟩, z⟩∣φ} ↔ ⟨⟨A, B⟩, C⟩ ∈ {⟨⟨x, y⟩, z⟩∣φ}))
20		df-oprab 3004	. . . . . . . . 9 ⊢ {⟨⟨x, y⟩, z⟩∣φ} = {w∣∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ)}
21	20	eleq2i 1153	. . . . . . . 8 ⊢ (w ∈ {⟨⟨x, y⟩, z⟩∣φ} ↔ w ∈ {w∣∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ)})
22		abid 1094	. . . . . . . 8 ⊢ (w ∈ {w∣∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ)} ↔ ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ))
23	21, 22	bitr2 152	. . . . . . 7 ⊢ (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ w ∈ {⟨⟨x, y⟩, z⟩∣φ})
24	19, 23	syl5bb 410	. . . . . 6 ⊢ (w = ⟨⟨A, B⟩, C⟩ → (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ ⟨⟨A, B⟩, C⟩ ∈ {⟨⟨x, y⟩, z⟩∣φ}))
25	24	adantl 305	. . . . 5 ⊢ (((A ∈ D ∧ B ∈ R ∧ C ∈ S) ∧ w = ⟨⟨A, B⟩, C⟩) → (∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ ⟨⟨A, B⟩, C⟩ ∈ {⟨⟨x, y⟩, z⟩∣φ}))
26		elex 1356	. . . . . . . . . 10 ⊢ (A ∈ D → ∃x x = A)
27		elex 1356	. . . . . . . . . 10 ⊢ (B ∈ R → ∃y y = B)
28		elex 1356	. . . . . . . . . 10 ⊢ (C ∈ S → ∃z z = C)
29	26, 27, 28	im3an 605	. . . . . . . . 9 ⊢ ((A ∈ D ∧ B ∈ R ∧ C ∈ S) → (∃x x = A ∧ ∃y y = B ∧ ∃z z = C))
30		eeeanv 981	. . . . . . . . 9 ⊢ (∃x∃y∃z(x = A ∧ y = B ∧ z = C) ↔ (∃x x = A ∧ ∃y y = B ∧ ∃z z = C))
31	29, 30	sylibr 175	. . . . . . . 8 ⊢ ((A ∈ D ∧ B ∈ R ∧ C ∈ S) → ∃x∃y∃z(x = A ∧ y = B ∧ z = C))
32	31	biantrurd 546	. . . . . . 7 ⊢ ((A ∈ D ∧ B ∈ R ∧ C ∈ S) → (θ ↔ (∃x∃y∃z(x = A ∧ y = B ∧ z = C) ∧ θ)))
33		19.41vvv 965	. . . . . . 7 ⊢ (∃x∃y∃z((x = A ∧ y = B ∧ z = C) ∧ θ) ↔ (∃x∃y∃z(x = A ∧ y = B ∧ z = C) ∧ θ))
34	32, 33	syl6rbbr 417	. . . . . 6 ⊢ ((A ∈ D ∧ B ∈ R ∧ C ∈ S) → (∃x∃y∃z((x = A ∧ y = B ∧ z = C) ∧ θ) ↔ θ))
35	34	adantr 306	. . . . 5 ⊢ (((A ∈ D ∧ B ∈ R ∧ C ∈ S) ∧ w = ⟨⟨A, B⟩, C⟩) → (∃x∃y∃z((x = A ∧ y = B ∧ z = C) ∧ θ) ↔ θ))
36	18, 25, 35	3bitr3d 423	. . . 4 ⊢ (((A ∈ D ∧ B ∈ R ∧ C ∈ S) ∧ w = ⟨⟨A, B⟩, C⟩) → (⟨⟨A, B⟩, C⟩ ∈ {⟨⟨x, y⟩, z⟩∣φ} ↔ θ))
37	36	exp 291	. . 3 ⊢ ((A ∈ D ∧ B ∈ R ∧ C ∈ S) → (w = ⟨⟨A, B⟩, C⟩ → (⟨⟨A, B⟩, C⟩ ∈ {⟨⟨x, y⟩, z⟩∣φ} ↔ θ)))
38	37	com12 13	. 2 ⊢ (w = ⟨⟨A, B⟩, C⟩ → ((A ∈ D ∧ B ∈ R ∧ C ∈ S) → (⟨⟨A, B⟩, C⟩ ∈ {⟨⟨x, y⟩, z⟩∣φ} ↔ θ)))
39	1, 38	vtocle 1391	1 ⊢ ((A ∈ D ∧ B ∈ R ∧ C ∈ S) → (⟨⟨A, B⟩, C⟩ ∈ {⟨⟨x, y⟩, z⟩∣φ} ↔ θ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   ∧ w3a 581  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  {copab2 3002

This theorem is referenced by:  oprabval 3047  oprabvalig 3048

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-oprab 3004

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