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| Mirrors > Home > MPE Home > Th. List > Mathboxes > idsset | Structured version Visualization version GIF version | ||
| Description: I is equal to the intersection of SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.) |
| Ref | Expression |
|---|---|
| idsset | ⊢ I = ( SSet ∩ ◡ SSet ) |
| 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 ) |
| Colors of variables: wff setvar class |
| 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) |
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