Theorem cnviinm 5278

Description: The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.)

Assertion

Ref	Expression
cnviinm	⊢ (∃𝑦 𝑦 ∈ 𝐴 → ◡∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 ◡𝐵)

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem cnviinm

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

Step	Hyp	Ref	Expression
1		eleq1w 2292	. . 3 ⊢ (𝑦 = 𝑎 → (𝑦 ∈ 𝐴 ↔ 𝑎 ∈ 𝐴))
2	1	cbvexv 1967	. 2 ⊢ (∃𝑦 𝑦 ∈ 𝐴 ↔ ∃𝑎 𝑎 ∈ 𝐴)
3		eleq1w 2292	. . . 4 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ 𝐴 ↔ 𝑎 ∈ 𝐴))
4	3	cbvexv 1967	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑎 𝑎 ∈ 𝐴)
5		relcnv 5114	. . . 4 ⊢ Rel ◡∩ 𝑥 ∈ 𝐴 𝐵
6		r19.2m 3581	. . . . . . . 8 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V)) → ∃𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V))
7	6	expcom 116	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V) → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V)))
8		relcnv 5114	. . . . . . . . 9 ⊢ Rel ◡𝐵
9		df-rel 4732	. . . . . . . . 9 ⊢ (Rel ◡𝐵 ↔ ◡𝐵 ⊆ (V × V))
10	8, 9	mpbi 145	. . . . . . . 8 ⊢ ◡𝐵 ⊆ (V × V)
11	10	a1i 9	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ◡𝐵 ⊆ (V × V))
12	7, 11	mprg 2589	. . . . . 6 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V))
13		iinss 4022	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V) → ∩ 𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V))
14	12, 13	syl 14	. . . . 5 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V))
15		df-rel 4732	. . . . 5 ⊢ (Rel ∩ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∩ 𝑥 ∈ 𝐴 ◡𝐵 ⊆ (V × V))
16	14, 15	sylibr 134	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → Rel ∩ 𝑥 ∈ 𝐴 ◡𝐵)
17		vex 2805	. . . . . . . 8 ⊢ 𝑏 ∈ V
18		vex 2805	. . . . . . . 8 ⊢ 𝑎 ∈ V
19	17, 18	opex 4321	. . . . . . 7 ⊢ ⟨𝑏, 𝑎⟩ ∈ V
20		eliin 3975	. . . . . . 7 ⊢ (⟨𝑏, 𝑎⟩ ∈ V → (⟨𝑏, 𝑎⟩ ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑏, 𝑎⟩ ∈ 𝐵))
21	19, 20	ax-mp 5	. . . . . 6 ⊢ (⟨𝑏, 𝑎⟩ ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑏, 𝑎⟩ ∈ 𝐵)
22	18, 17	opelcnv 4912	. . . . . 6 ⊢ (⟨𝑎, 𝑏⟩ ∈ ◡∩ 𝑥 ∈ 𝐴 𝐵 ↔ ⟨𝑏, 𝑎⟩ ∈ ∩ 𝑥 ∈ 𝐴 𝐵)
23	18, 17	opex 4321	. . . . . . . 8 ⊢ ⟨𝑎, 𝑏⟩ ∈ V
24		eliin 3975	. . . . . . . 8 ⊢ (⟨𝑎, 𝑏⟩ ∈ V → (⟨𝑎, 𝑏⟩ ∈ ∩ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑎, 𝑏⟩ ∈ ◡𝐵))
25	23, 24	ax-mp 5	. . . . . . 7 ⊢ (⟨𝑎, 𝑏⟩ ∈ ∩ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑎, 𝑏⟩ ∈ ◡𝐵)
26	18, 17	opelcnv 4912	. . . . . . . 8 ⊢ (⟨𝑎, 𝑏⟩ ∈ ◡𝐵 ↔ ⟨𝑏, 𝑎⟩ ∈ 𝐵)
27	26	ralbii 2538	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ⟨𝑎, 𝑏⟩ ∈ ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑏, 𝑎⟩ ∈ 𝐵)
28	25, 27	bitri 184	. . . . . 6 ⊢ (⟨𝑎, 𝑏⟩ ∈ ∩ 𝑥 ∈ 𝐴 ◡𝐵 ↔ ∀𝑥 ∈ 𝐴 ⟨𝑏, 𝑎⟩ ∈ 𝐵)
29	21, 22, 28	3bitr4i 212	. . . . 5 ⊢ (⟨𝑎, 𝑏⟩ ∈ ◡∩ 𝑥 ∈ 𝐴 𝐵 ↔ ⟨𝑎, 𝑏⟩ ∈ ∩ 𝑥 ∈ 𝐴 ◡𝐵)
30	29	eqrelriv 4819	. . . 4 ⊢ ((Rel ◡∩ 𝑥 ∈ 𝐴 𝐵 ∧ Rel ∩ 𝑥 ∈ 𝐴 ◡𝐵) → ◡∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 ◡𝐵)
31	5, 16, 30	sylancr 414	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ◡∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 ◡𝐵)
32	4, 31	sylbir 135	. 2 ⊢ (∃𝑎 𝑎 ∈ 𝐴 → ◡∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 ◡𝐵)
33	2, 32	sylbi 121	1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ◡∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 ◡𝐵)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511  Vcvv 2802   ⊆ wss 3200  ⟨cop 3672  ∩ ciin 3971   × cxp 4723  ^◡ccnv 4724  Rel wrel 4730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-iin 3973  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733

This theorem is referenced by: (None)