Theorem cnviin 6286

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 6104	. 2 ⊢ Rel ◡∩ 𝑥 ∈ 𝐴 𝐵
2		relcnv 6104	. . . . . . 7 ⊢ Rel ◡𝐵
3		df-rel 5684	. . . . . . 7 ⊢ (Rel ◡𝐵 ↔ ◡𝐵 ⊆ (V × V))
4	2, 3	mpbi 229	. . . . . 6 ⊢ ◡𝐵 ⊆ (V × V)
5	4	rgenw 3066	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V)
6		r19.2z 4495	. . . . 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 5060	. . . 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 5684	. . 3 ⊢ (Rel ∩ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∩ 𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V))
11	9, 10	sylibr 233	. 2 ⊢ (𝐴 ≠ ∅ → Rel ∩ 𝑥 ∈ 𝐴 ◡𝐵)
12		opex 5465	. . . . 5 ⊢ ⟨𝑏, 𝑎⟩ ∈ V
13		eliin 5003	. . . . 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 3479	. . . . 5 ⊢ 𝑎 ∈ V
16		vex 3479	. . . . 5 ⊢ 𝑏 ∈ V
17	15, 16	opelcnv 5882	. . . 4 ⊢ (⟨𝑎, 𝑏⟩ ∈ ◡∩ 𝑥 ∈ 𝐴 𝐵 ↔ ⟨𝑏, 𝑎⟩ ∈ ∩ 𝑥 ∈ 𝐴 𝐵)
18		opex 5465	. . . . . 6 ⊢ ⟨𝑎, 𝑏⟩ ∈ V
19		eliin 5003	. . . . . 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 5882	. . . . . 6 ⊢ (⟨𝑎, 𝑏⟩ ∈ ◡𝐵 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝐵)
22	21	ralbii 3094	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ⟨𝑎, 𝑏⟩ ∈ ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑏, 𝑎⟩ ∈ 𝐵)
23	20, 22	bitri 275	. . . 4 ⊢ (⟨𝑎, 𝑏⟩ ∈ ∩ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑏, 𝑎⟩ ∈ 𝐵)
24	14, 17, 23	3bitr4i 303	. . 3 ⊢ (⟨𝑎, 𝑏⟩ ∈ ◡∩ 𝑥 ∈ 𝐴 𝐵 ↔ ⟨𝑎, 𝑏⟩ ∈ ∩ 𝑥 ∈ 𝐴 ◡𝐵)
25	24	eqrelriv 5790	. 2 ⊢ ((Rel ◡∩ 𝑥 ∈ 𝐴 𝐵 ∧ Rel ∩ 𝑥 ∈ 𝐴 ◡𝐵) → ◡∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 ◡𝐵)
26	1, 11, 25	sylancr 588	1 ⊢ (𝐴 ≠ ∅ → ◡∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 ◡𝐵)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  Vcvv 3475   ⊆ wss 3949  ∅c0 4323  ⟨cop 4635  ∩ ciin 4999   × cxp 5675  ^◡ccnv 5676  Rel wrel 5682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-iin 5001  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685

This theorem is referenced by: (None)