Theorem cnviin 6119

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 5943	. 2 ⊢ Rel ◡∩ 𝑥 ∈ 𝐴 𝐵
2		relcnv 5943	. . . . . . 7 ⊢ Rel ◡𝐵
3		df-rel 5534	. . . . . . 7 ⊢ (Rel ◡𝐵 ↔ ◡𝐵 ⊆ (V × V))
4	2, 3	mpbi 233	. . . . . 6 ⊢ ◡𝐵 ⊆ (V × V)
5	4	rgenw 3082	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V)
6		r19.2z 4391	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V)) → ∃𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V))
7	5, 6	mpan2 690	. . . 4 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V))
8		iinss 4948	. . . 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 5534	. . 3 ⊢ (Rel ∩ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∩ 𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V))
11	9, 10	sylibr 237	. 2 ⊢ (𝐴 ≠ ∅ → Rel ∩ 𝑥 ∈ 𝐴 ◡𝐵)
12		opex 5327	. . . . 5 ⊢ ⟨𝑏, 𝑎⟩ ∈ V
13		eliin 4891	. . . . 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 3413	. . . . 5 ⊢ 𝑎 ∈ V
16		vex 3413	. . . . 5 ⊢ 𝑏 ∈ V
17	15, 16	opelcnv 5726	. . . 4 ⊢ (⟨𝑎, 𝑏⟩ ∈ ◡∩ 𝑥 ∈ 𝐴 𝐵 ↔ ⟨𝑏, 𝑎⟩ ∈ ∩ 𝑥 ∈ 𝐴 𝐵)
18		opex 5327	. . . . . 6 ⊢ ⟨𝑎, 𝑏⟩ ∈ V
19		eliin 4891	. . . . . 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 5726	. . . . . 6 ⊢ (⟨𝑎, 𝑏⟩ ∈ ◡𝐵 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝐵)
22	21	ralbii 3097	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ⟨𝑎, 𝑏⟩ ∈ ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑏, 𝑎⟩ ∈ 𝐵)
23	20, 22	bitri 278	. . . 4 ⊢ (⟨𝑎, 𝑏⟩ ∈ ∩ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑏, 𝑎⟩ ∈ 𝐵)
24	14, 17, 23	3bitr4i 306	. . 3 ⊢ (⟨𝑎, 𝑏⟩ ∈ ◡∩ 𝑥 ∈ 𝐴 𝐵 ↔ ⟨𝑎, 𝑏⟩ ∈ ∩ 𝑥 ∈ 𝐴 ◡𝐵)
25	24	eqrelriv 5635	. 2 ⊢ ((Rel ◡∩ 𝑥 ∈ 𝐴 𝐵 ∧ Rel ∩ 𝑥 ∈ 𝐴 ◡𝐵) → ◡∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 ◡𝐵)
26	1, 11, 25	sylancr 590	1 ⊢ (𝐴 ≠ ∅ → ◡∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 ◡𝐵)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ∃wrex 3071  Vcvv 3409   ⊆ wss 3860  ∅c0 4227  ⟨cop 4531  ∩ ciin 4887   × cxp 5525  ^◡ccnv 5526  Rel wrel 5532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ne 2952  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-iin 4889  df-br 5036  df-opab 5098  df-xp 5533  df-rel 5534  df-cnv 5535

This theorem is referenced by: (None)