Theorem idsset 35891

Description: I is equal to the intersection of SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
idsset	⊢ I = ( SSet ∩ ◡ SSet )

Proof of Theorem idsset

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reli 5836	. 2 ⊢ Rel I
2		relsset 35889	. . 3 ⊢ Rel SSet
3		relin1 5822	. . 3 ⊢ (Rel SSet → Rel ( SSet ∩ ◡ SSet ))
4	2, 3	ax-mp 5	. 2 ⊢ Rel ( SSet ∩ ◡ SSet )
5		eqss 3999	. . 3 ⊢ (𝑦 = 𝑧 ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦))
6		vex 3484	. . . 4 ⊢ 𝑧 ∈ V
7	6	ideq 5863	. . 3 ⊢ (𝑦 I 𝑧 ↔ 𝑦 = 𝑧)
8		brin 5195	. . . 4 ⊢ (𝑦( SSet ∩ ◡ SSet )𝑧 ↔ (𝑦 SSet 𝑧 ∧ 𝑦◡ SSet 𝑧))
9	6	brsset 35890	. . . . 5 ⊢ (𝑦 SSet 𝑧 ↔ 𝑦 ⊆ 𝑧)
10		vex 3484	. . . . . . 7 ⊢ 𝑦 ∈ V
11	10, 6	brcnv 5893	. . . . . 6 ⊢ (𝑦◡ SSet 𝑧 ↔ 𝑧 SSet 𝑦)
12	10	brsset 35890	. . . . . 6 ⊢ (𝑧 SSet 𝑦 ↔ 𝑧 ⊆ 𝑦)
13	11, 12	bitri 275	. . . . 5 ⊢ (𝑦◡ SSet 𝑧 ↔ 𝑧 ⊆ 𝑦)
14	9, 13	anbi12i 628	. . . 4 ⊢ ((𝑦 SSet 𝑧 ∧ 𝑦◡ SSet 𝑧) ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦))
15	8, 14	bitri 275	. . 3 ⊢ (𝑦( SSet ∩ ◡ SSet )𝑧 ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦))
16	5, 7, 15	3bitr4i 303	. 2 ⊢ (𝑦 I 𝑧 ↔ 𝑦( SSet ∩ ◡ SSet )𝑧)
17	1, 4, 16	eqbrriv 5801	1 ⊢ I = ( SSet ∩ ◡ SSet )

Syntax hints:   ∧ wa 395   = wceq 1540   ∩ cin 3950   ⊆ wss 3951   class class class wbr 5143   I cid 5577  ^◡ccnv 5684  Rel wrel 5690   SSet csset 35833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-eprel 5584  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-1st 8014  df-2nd 8015  df-txp 35855  df-sset 35857

This theorem is referenced by: (None)