Theorem cnvopab 2632

Description: The converse of a class abstraction of ordered pairs.

Assertion

Ref	Expression
cnvopab	⊢ ◡{⟨x, y⟩∣φ} = {⟨y, x⟩∣φ}

Distinct variable group(s):   x,y

Proof of Theorem cnvopab

Step	Hyp	Ref	Expression
1		relcnv 2624	. 2 ⊢ Rel ◡{⟨x, y⟩∣φ}
2		relopab 2494	. 2 ⊢ Rel {⟨y, x⟩∣φ}
3		visset 1350	. . . 4 ⊢ w ∈ V
4		visset 1350	. . . 4 ⊢ z ∈ V
5	3, 4	opelcnv 2518	. . 3 ⊢ (⟨w, z⟩ ∈ ◡{⟨x, y⟩∣φ} ↔ ⟨z, w⟩ ∈ {⟨x, y⟩∣φ})
6		ax-17 925	. . . . . 6 ⊢ (y ∈ ⟨z, w⟩ → ∀x y ∈ ⟨z, w⟩)
7		hbopab1 2112	. . . . . 6 ⊢ (z ∈ {⟨x, y⟩∣φ} → ∀x z ∈ {⟨x, y⟩∣φ})
8	6, 7	hbel 1172	. . . . 5 ⊢ (⟨z, w⟩ ∈ {⟨x, y⟩∣φ} → ∀x⟨z, w⟩ ∈ {⟨x, y⟩∣φ})
9		ax-17 925	. . . . . 6 ⊢ (y ∈ ⟨w, z⟩ → ∀x y ∈ ⟨w, z⟩)
10		hbopab2 2113	. . . . . 6 ⊢ (z ∈ {⟨y, x⟩∣φ} → ∀x z ∈ {⟨y, x⟩∣φ})
11	9, 10	hbel 1172	. . . . 5 ⊢ (⟨w, z⟩ ∈ {⟨y, x⟩∣φ} → ∀x⟨w, z⟩ ∈ {⟨y, x⟩∣φ})
12	8, 11	hbbi 705	. . . 4 ⊢ ((⟨z, w⟩ ∈ {⟨x, y⟩∣φ} ↔ ⟨w, z⟩ ∈ {⟨y, x⟩∣φ}) → ∀x(⟨z, w⟩ ∈ {⟨x, y⟩∣φ} ↔ ⟨w, z⟩ ∈ {⟨y, x⟩∣φ}))
13		opeq1 1876	. . . . . 6 ⊢ (x = z → ⟨x, w⟩ = ⟨z, w⟩)
14	13	eleq1d 1155	. . . . 5 ⊢ (x = z → (⟨x, w⟩ ∈ {⟨x, y⟩∣φ} ↔ ⟨z, w⟩ ∈ {⟨x, y⟩∣φ}))
15		opeq2 1877	. . . . . 6 ⊢ (x = z → ⟨w, x⟩ = ⟨w, z⟩)
16	15	eleq1d 1155	. . . . 5 ⊢ (x = z → (⟨w, x⟩ ∈ {⟨y, x⟩∣φ} ↔ ⟨w, z⟩ ∈ {⟨y, x⟩∣φ}))
17	14, 16	bibi12d 477	. . . 4 ⊢ (x = z → ((⟨x, w⟩ ∈ {⟨x, y⟩∣φ} ↔ ⟨w, x⟩ ∈ {⟨y, x⟩∣φ}) ↔ (⟨z, w⟩ ∈ {⟨x, y⟩∣φ} ↔ ⟨w, z⟩ ∈ {⟨y, x⟩∣φ})))
18		ax-17 925	. . . . . . 7 ⊢ (z ∈ ⟨x, w⟩ → ∀y z ∈ ⟨x, w⟩)
19		hbopab2 2113	. . . . . . 7 ⊢ (z ∈ {⟨x, y⟩∣φ} → ∀y z ∈ {⟨x, y⟩∣φ})
20	18, 19	hbel 1172	. . . . . 6 ⊢ (⟨x, w⟩ ∈ {⟨x, y⟩∣φ} → ∀y⟨x, w⟩ ∈ {⟨x, y⟩∣φ})
21		ax-17 925	. . . . . . 7 ⊢ (z ∈ ⟨w, x⟩ → ∀y z ∈ ⟨w, x⟩)
22		hbopab1 2112	. . . . . . 7 ⊢ (z ∈ {⟨y, x⟩∣φ} → ∀y z ∈ {⟨y, x⟩∣φ})
23	21, 22	hbel 1172	. . . . . 6 ⊢ (⟨w, x⟩ ∈ {⟨y, x⟩∣φ} → ∀y⟨w, x⟩ ∈ {⟨y, x⟩∣φ})
24	20, 23	hbbi 705	. . . . 5 ⊢ ((⟨x, w⟩ ∈ {⟨x, y⟩∣φ} ↔ ⟨w, x⟩ ∈ {⟨y, x⟩∣φ}) → ∀y(⟨x, w⟩ ∈ {⟨x, y⟩∣φ} ↔ ⟨w, x⟩ ∈ {⟨y, x⟩∣φ}))
25		opeq2 1877	. . . . . . 7 ⊢ (y = w → ⟨x, y⟩ = ⟨x, w⟩)
26	25	eleq1d 1155	. . . . . 6 ⊢ (y = w → (⟨x, y⟩ ∈ {⟨x, y⟩∣φ} ↔ ⟨x, w⟩ ∈ {⟨x, y⟩∣φ}))
27		opeq1 1876	. . . . . . 7 ⊢ (y = w → ⟨y, x⟩ = ⟨w, x⟩)
28	27	eleq1d 1155	. . . . . 6 ⊢ (y = w → (⟨y, x⟩ ∈ {⟨y, x⟩∣φ} ↔ ⟨w, x⟩ ∈ {⟨y, x⟩∣φ}))
29	26, 28	bibi12d 477	. . . . 5 ⊢ (y = w → ((⟨x, y⟩ ∈ {⟨x, y⟩∣φ} ↔ ⟨y, x⟩ ∈ {⟨y, x⟩∣φ}) ↔ (⟨x, w⟩ ∈ {⟨x, y⟩∣φ} ↔ ⟨w, x⟩ ∈ {⟨y, x⟩∣φ})))
30		opabid 2099	. . . . . 6 ⊢ (⟨x, y⟩ ∈ {⟨x, y⟩∣φ} ↔ φ)
31		opabid 2099	. . . . . 6 ⊢ (⟨y, x⟩ ∈ {⟨y, x⟩∣φ} ↔ φ)
32	30, 31	bitr4 154	. . . . 5 ⊢ (⟨x, y⟩ ∈ {⟨x, y⟩∣φ} ↔ ⟨y, x⟩ ∈ {⟨y, x⟩∣φ})
33	24, 29, 32	chv2 850	. . . 4 ⊢ (⟨x, w⟩ ∈ {⟨x, y⟩∣φ} ↔ ⟨w, x⟩ ∈ {⟨y, x⟩∣φ})
34	12, 17, 33	chv2 850	. . 3 ⊢ (⟨z, w⟩ ∈ {⟨x, y⟩∣φ} ↔ ⟨w, z⟩ ∈ {⟨y, x⟩∣φ})
35	5, 34	bitr 151	. 2 ⊢ (⟨w, z⟩ ∈ ◡{⟨x, y⟩∣φ} ↔ ⟨w, z⟩ ∈ {⟨y, x⟩∣φ})
36	1, 2, 35	cleqreli 2484	1 ⊢ ◡{⟨x, y⟩∣φ} = {⟨y, x⟩∣φ}

Syntax hints:   ↔ wb 127   = weq 797   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  {copab 2055  ^◡ccnv 2409

This theorem is referenced by:  en2d 3303

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-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426

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