Theorem idsset 34171

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 5733	. 2 ⊢ Rel I
2		relsset 34169	. . 3 ⊢ Rel SSet
3		relin1 5719	. . 3 ⊢ (Rel SSet → Rel ( SSet ∩ ◡ SSet ))
4	2, 3	ax-mp 5	. 2 ⊢ Rel ( SSet ∩ ◡ SSet )
5		eqss 3940	. . 3 ⊢ (𝑦 = 𝑧 ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦))
6		vex 3434	. . . 4 ⊢ 𝑧 ∈ V
7	6	ideq 5758	. . 3 ⊢ (𝑦 I 𝑧 ↔ 𝑦 = 𝑧)
8		brin 5130	. . . 4 ⊢ (𝑦( SSet ∩ ◡ SSet )𝑧 ↔ (𝑦 SSet 𝑧 ∧ 𝑦◡ SSet 𝑧))
9	6	brsset 34170	. . . . 5 ⊢ (𝑦 SSet 𝑧 ↔ 𝑦 ⊆ 𝑧)
10		vex 3434	. . . . . . 7 ⊢ 𝑦 ∈ V
11	10, 6	brcnv 5788	. . . . . 6 ⊢ (𝑦◡ SSet 𝑧 ↔ 𝑧 SSet 𝑦)
12	10	brsset 34170	. . . . . 6 ⊢ (𝑧 SSet 𝑦 ↔ 𝑧 ⊆ 𝑦)
13	11, 12	bitri 274	. . . . 5 ⊢ (𝑦◡ SSet 𝑧 ↔ 𝑧 ⊆ 𝑦)
14	9, 13	anbi12i 626	. . . 4 ⊢ ((𝑦 SSet 𝑧 ∧ 𝑦◡ SSet 𝑧) ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦))
15	8, 14	bitri 274	. . 3 ⊢ (𝑦( SSet ∩ ◡ SSet )𝑧 ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦))
16	5, 7, 15	3bitr4i 302	. 2 ⊢ (𝑦 I 𝑧 ↔ 𝑦( SSet ∩ ◡ SSet )𝑧)
17	1, 4, 16	eqbrriv 5698	1 ⊢ I = ( SSet ∩ ◡ SSet )

Syntax hints:   ∧ wa 395   = wceq 1541   ∩ cin 3890   ⊆ wss 3891   class class class wbr 5078   I cid 5487  ^◡ccnv 5587  Rel wrel 5593   SSet csset 34113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-eprel 5494  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-fo 6436  df-fv 6438  df-1st 7817  df-2nd 7818  df-txp 34135  df-sset 34137

This theorem is referenced by: (None)