Theorem idsset 31167

Description: I is equal to 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 5171	. 2 ⊢ Rel I
2		relsset 31165	. . 3 ⊢ Rel SSet
3		relin1 5159	. . 3 ⊢ (Rel SSet → Rel ( SSet ∩ ◡ SSet ))
4	2, 3	ax-mp 5	. 2 ⊢ Rel ( SSet ∩ ◡ SSet )
5		eqss 3583	. . 3 ⊢ (𝑦 = 𝑧 ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦))
6		vex 3176	. . . 4 ⊢ 𝑧 ∈ V
7	6	ideq 5196	. . 3 ⊢ (𝑦 I 𝑧 ↔ 𝑦 = 𝑧)
8		brin 4634	. . . 4 ⊢ (𝑦( SSet ∩ ◡ SSet )𝑧 ↔ (𝑦 SSet 𝑧 ∧ 𝑦◡ SSet 𝑧))
9	6	brsset 31166	. . . . 5 ⊢ (𝑦 SSet 𝑧 ↔ 𝑦 ⊆ 𝑧)
10		vex 3176	. . . . . . 7 ⊢ 𝑦 ∈ V
11	10, 6	brcnv 5227	. . . . . 6 ⊢ (𝑦◡ SSet 𝑧 ↔ 𝑧 SSet 𝑦)
12	10	brsset 31166	. . . . . 6 ⊢ (𝑧 SSet 𝑦 ↔ 𝑧 ⊆ 𝑦)
13	11, 12	bitri 263	. . . . 5 ⊢ (𝑦◡ SSet 𝑧 ↔ 𝑧 ⊆ 𝑦)
14	9, 13	anbi12i 729	. . . 4 ⊢ ((𝑦 SSet 𝑧 ∧ 𝑦◡ SSet 𝑧) ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦))
15	8, 14	bitri 263	. . 3 ⊢ (𝑦( SSet ∩ ◡ SSet )𝑧 ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦))
16	5, 7, 15	3bitr4i 291	. 2 ⊢ (𝑦 I 𝑧 ↔ 𝑦( SSet ∩ ◡ SSet )𝑧)
17	1, 4, 16	eqbrriv 5138	1 ⊢ I = ( SSet ∩ ◡ SSet )

Syntax hints:   ∧ wa 383   = wceq 1475   ∩ cin 3539   ⊆ wss 3540   class class class wbr 4583   I cid 4948  ^◡ccnv 5037  Rel wrel 5043   SSet csset 31108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-eprel 4949  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-1st 7059  df-2nd 7060  df-txp 31130  df-sset 31132

This theorem is referenced by: (None)