Theorem cnviin 6275

Description: The converse of an intersection is the intersection of the converse. (Contributed by FL, 15-Oct-2012.)

Assertion

Ref	Expression
cnviin	⊢ (𝐴 ≠ ∅ → ◡∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 ◡𝐵)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem cnviin

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcnv 6095	. 2 ⊢ Rel ◡∩ 𝑥 ∈ 𝐴 𝐵
2		relcnv 6095	. . . . . . 7 ⊢ Rel ◡𝐵
3		df-rel 5656	. . . . . . 7 ⊢ (Rel ◡𝐵 ↔ ◡𝐵 ⊆ (V × V))
4	2, 3	mpbi 232	. . . . . 6 ⊢ ◡𝐵 ⊆ (V × V)
5	4	rgenw 3082	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V)
6		r19.2z 4455	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V)) → ∃𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V))
7	5, 6	mpan2 701	. . . 4 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V))
8		iinss 5016	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V) → ∩ 𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V))
9	7, 8	syl 17	. . 3 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V))
10		df-rel 5656	. . 3 ⊢ (Rel ∩ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∩ 𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V))
11	9, 10	sylibr 236	. 2 ⊢ (𝐴 ≠ ∅ → Rel ∩ 𝑥 ∈ 𝐴 ◡𝐵)
12		opex 5433	. . . . 5 ⊢ ⟨𝑏, 𝑎⟩ ∈ V
13		eliin 4956	. . . . 5 ⊢ (⟨𝑏, 𝑎⟩ ∈ V → (⟨𝑏, 𝑎⟩ ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑏, 𝑎⟩ ∈ 𝐵))
14	12, 13	ax-mp 5	. . . 4 ⊢ (⟨𝑏, 𝑎⟩ ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑏, 𝑎⟩ ∈ 𝐵)
15		vex 3460	. . . . 5 ⊢ 𝑎 ∈ V
16		vex 3460	. . . . 5 ⊢ 𝑏 ∈ V
17	15, 16	opelcnv 5855	. . . 4 ⊢ (⟨𝑎, 𝑏⟩ ∈ ◡∩ 𝑥 ∈ 𝐴 𝐵 ↔ ⟨𝑏, 𝑎⟩ ∈ ∩ 𝑥 ∈ 𝐴 𝐵)
18		opex 5433	. . . . . 6 ⊢ ⟨𝑎, 𝑏⟩ ∈ V
19		eliin 4956	. . . . . 6 ⊢ (⟨𝑎, 𝑏⟩ ∈ V → (⟨𝑎, 𝑏⟩ ∈ ∩ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑎, 𝑏⟩ ∈ ◡𝐵))
20	18, 19	ax-mp 5	. . . . 5 ⊢ (⟨𝑎, 𝑏⟩ ∈ ∩ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑎, 𝑏⟩ ∈ ◡𝐵)
21	15, 16	opelcnv 5855	. . . . . 6 ⊢ (⟨𝑎, 𝑏⟩ ∈ ◡𝐵 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝐵)
22	21	ralbii 3110	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ⟨𝑎, 𝑏⟩ ∈ ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑏, 𝑎⟩ ∈ 𝐵)
23	20, 22	bitri 277	. . . 4 ⊢ (⟨𝑎, 𝑏⟩ ∈ ∩ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑏, 𝑎⟩ ∈ 𝐵)
24	14, 17, 23	3bitr4i 305	. . 3 ⊢ (⟨𝑎, 𝑏⟩ ∈ ◡∩ 𝑥 ∈ 𝐴 𝐵 ↔ ⟨𝑎, 𝑏⟩ ∈ ∩ 𝑥 ∈ 𝐴 ◡𝐵)
25	24	eqrelriv 5763	. 2 ⊢ ((Rel ◡∩ 𝑥 ∈ 𝐴 𝐵 ∧ Rel ∩ 𝑥 ∈ 𝐴 ◡𝐵) → ◡∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 ◡𝐵)
26	1, 11, 25	sylancr 596	1 ⊢ (𝐴 ≠ ∅ → ◡∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 ◡𝐵)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1562   ∈ wcel 2144   ≠ wne 2959  ∀wral 3078  ∃wrex 3088  Vcvv 3456   ⊆ wss 3906  ∅c0 4287  ⟨cop 4590  ∩ ciin 4952   × cxp 5647  ^◡ccnv 5648  Rel wrel 5654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-iin 4954  df-br 5103  df-opab 5165  df-xp 5655  df-rel 5656  df-cnv 5657

This theorem is referenced by: (None)