Theorem ssopab2 2119

Description: Equivalence of abstraction subclass and implication.

Assertion

Ref	Expression
ssopab2	⊢ ({⟨x, y⟩∣φ} ⊆ {⟨x, y⟩∣ψ} ↔ ∀x∀y(φ → ψ))

Distinct variable group(s):   x,y

Proof of Theorem ssopab2

Step	Hyp	Ref	Expression
1		hbopab1 2112	. . . 4 ⊢ (z ∈ {⟨x, y⟩∣φ} → ∀x z ∈ {⟨x, y⟩∣φ})
2		hbopab1 2112	. . . 4 ⊢ (z ∈ {⟨x, y⟩∣ψ} → ∀x z ∈ {⟨x, y⟩∣ψ})
3	1, 2	hbss 1501	. . 3 ⊢ ({⟨x, y⟩∣φ} ⊆ {⟨x, y⟩∣ψ} → ∀x{⟨x, y⟩∣φ} ⊆ {⟨x, y⟩∣ψ})
4		hbopab2 2113	. . . . 5 ⊢ (z ∈ {⟨x, y⟩∣φ} → ∀y z ∈ {⟨x, y⟩∣φ})
5		hbopab2 2113	. . . . 5 ⊢ (z ∈ {⟨x, y⟩∣ψ} → ∀y z ∈ {⟨x, y⟩∣ψ})
6	4, 5	hbss 1501	. . . 4 ⊢ ({⟨x, y⟩∣φ} ⊆ {⟨x, y⟩∣ψ} → ∀y{⟨x, y⟩∣φ} ⊆ {⟨x, y⟩∣ψ})
7		opex 1893	. . . . . 6 ⊢ ⟨x, y⟩ ∈ V
8	7	isseti 1352	. . . . 5 ⊢ ∃z z = ⟨x, y⟩
9		copsexg 1902	. . . . . . . . 9 ⊢ (z = ⟨x, y⟩ → (φ ↔ ∃x∃y(z = ⟨x, y⟩ ∧ φ)))
10		copsexg 1902	. . . . . . . . 9 ⊢ (z = ⟨x, y⟩ → (ψ ↔ ∃x∃y(z = ⟨x, y⟩ ∧ ψ)))
11	9, 10	imbi12d 474	. . . . . . . 8 ⊢ (z = ⟨x, y⟩ → ((φ → ψ) ↔ (∃x∃y(z = ⟨x, y⟩ ∧ φ) → ∃x∃y(z = ⟨x, y⟩ ∧ ψ))))
12		ss2ab 1551	. . . . . . . . 9 ⊢ ({z∣∃x∃y(z = ⟨x, y⟩ ∧ φ)} ⊆ {z∣∃x∃y(z = ⟨x, y⟩ ∧ ψ)} ↔ ∀z(∃x∃y(z = ⟨x, y⟩ ∧ φ) → ∃x∃y(z = ⟨x, y⟩ ∧ ψ)))
13		ax-4 673	. . . . . . . . 9 ⊢ (∀z(∃x∃y(z = ⟨x, y⟩ ∧ φ) → ∃x∃y(z = ⟨x, y⟩ ∧ ψ)) → (∃x∃y(z = ⟨x, y⟩ ∧ φ) → ∃x∃y(z = ⟨x, y⟩ ∧ ψ)))
14	12, 13	sylbi 174	. . . . . . . 8 ⊢ ({z∣∃x∃y(z = ⟨x, y⟩ ∧ φ)} ⊆ {z∣∃x∃y(z = ⟨x, y⟩ ∧ ψ)} → (∃x∃y(z = ⟨x, y⟩ ∧ φ) → ∃x∃y(z = ⟨x, y⟩ ∧ ψ)))
15	11, 14	syl5bir 184	. . . . . . 7 ⊢ (z = ⟨x, y⟩ → ({z∣∃x∃y(z = ⟨x, y⟩ ∧ φ)} ⊆ {z∣∃x∃y(z = ⟨x, y⟩ ∧ ψ)} → (φ → ψ)))
16		df-opab 2098	. . . . . . . 8 ⊢ {⟨x, y⟩∣φ} = {z∣∃x∃y(z = ⟨x, y⟩ ∧ φ)}
17		df-opab 2098	. . . . . . . 8 ⊢ {⟨x, y⟩∣ψ} = {z∣∃x∃y(z = ⟨x, y⟩ ∧ ψ)}
18	16, 17	sseq12i 1526	. . . . . . 7 ⊢ ({⟨x, y⟩∣φ} ⊆ {⟨x, y⟩∣ψ} ↔ {z∣∃x∃y(z = ⟨x, y⟩ ∧ φ)} ⊆ {z∣∃x∃y(z = ⟨x, y⟩ ∧ ψ)})
19	15, 18	syl5ib 181	. . . . . 6 ⊢ (z = ⟨x, y⟩ → ({⟨x, y⟩∣φ} ⊆ {⟨x, y⟩∣ψ} → (φ → ψ)))
20	19	19.23aiv 952	. . . . 5 ⊢ (∃z z = ⟨x, y⟩ → ({⟨x, y⟩∣φ} ⊆ {⟨x, y⟩∣ψ} → (φ → ψ)))
21	8, 20	ax-mp 6	. . . 4 ⊢ ({⟨x, y⟩∣φ} ⊆ {⟨x, y⟩∣ψ} → (φ → ψ))
22	6, 21	19.21ai 740	. . 3 ⊢ ({⟨x, y⟩∣φ} ⊆ {⟨x, y⟩∣ψ} → ∀y(φ → ψ))
23	3, 22	19.21ai 740	. 2 ⊢ ({⟨x, y⟩∣φ} ⊆ {⟨x, y⟩∣ψ} → ∀x∀y(φ → ψ))
24		hba1 698	. . . . . 6 ⊢ (∀x∀y(φ → ψ) → ∀x∀x∀y(φ → ψ))
25		hba1 698	. . . . . . . 8 ⊢ (∀y(φ → ψ) → ∀y∀y(φ → ψ))
26		ax-4 673	. . . . . . . . 9 ⊢ (∀y(φ → ψ) → (φ → ψ))
27	26	anim2d 433	. . . . . . . 8 ⊢ (∀y(φ → ψ) → ((z = ⟨x, y⟩ ∧ φ) → (z = ⟨x, y⟩ ∧ ψ)))
28	25, 27	19.22d 744	. . . . . . 7 ⊢ (∀y(φ → ψ) → (∃y(z = ⟨x, y⟩ ∧ φ) → ∃y(z = ⟨x, y⟩ ∧ ψ)))
29	28	a4s 682	. . . . . 6 ⊢ (∀x∀y(φ → ψ) → (∃y(z = ⟨x, y⟩ ∧ φ) → ∃y(z = ⟨x, y⟩ ∧ ψ)))
30	24, 29	19.22d 744	. . . . 5 ⊢ (∀x∀y(φ → ψ) → (∃x∃y(z = ⟨x, y⟩ ∧ φ) → ∃x∃y(z = ⟨x, y⟩ ∧ ψ)))
31	30	19.21aiv 943	. . . 4 ⊢ (∀x∀y(φ → ψ) → ∀z(∃x∃y(z = ⟨x, y⟩ ∧ φ) → ∃x∃y(z = ⟨x, y⟩ ∧ ψ)))
32	31, 12	sylibr 175	. . 3 ⊢ (∀x∀y(φ → ψ) → {z∣∃x∃y(z = ⟨x, y⟩ ∧ φ)} ⊆ {z∣∃x∃y(z = ⟨x, y⟩ ∧ ψ)})
33	32, 16, 17	3sstr4g 1541	. 2 ⊢ (∀x∀y(φ → ψ) → {⟨x, y⟩∣φ} ⊆ {⟨x, y⟩∣ψ})
34	23, 33	impbi 139	1 ⊢ ({⟨x, y⟩∣φ} ⊆ {⟨x, y⟩∣ψ} ↔ ∀x∀y(φ → ψ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678  {cab 1090   = wceq 1091   ⊆ wss 1487  ⟨cop 1810  {copab 2055

This theorem is referenced by:  ssopab2i 2120  cnvss 2512  cotr 2625  cnvsym 2626  dffun2 2674

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-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-opab 2098

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