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Mathbox for Scott Fenton |
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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 5495 | . 2 ⊢ Rel I | |
2 | relsset 32598 | . . 3 ⊢ Rel SSet | |
3 | relin1 5483 | . . 3 ⊢ (Rel SSet → Rel ( SSet ∩ ◡ SSet )) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ Rel ( SSet ∩ ◡ SSet ) |
5 | eqss 3835 | . . 3 ⊢ (𝑦 = 𝑧 ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦)) | |
6 | vex 3400 | . . . 4 ⊢ 𝑧 ∈ V | |
7 | 6 | ideq 5520 | . . 3 ⊢ (𝑦 I 𝑧 ↔ 𝑦 = 𝑧) |
8 | brin 4938 | . . . 4 ⊢ (𝑦( SSet ∩ ◡ SSet )𝑧 ↔ (𝑦 SSet 𝑧 ∧ 𝑦◡ SSet 𝑧)) | |
9 | 6 | brsset 32599 | . . . . 5 ⊢ (𝑦 SSet 𝑧 ↔ 𝑦 ⊆ 𝑧) |
10 | vex 3400 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
11 | 10, 6 | brcnv 5550 | . . . . . 6 ⊢ (𝑦◡ SSet 𝑧 ↔ 𝑧 SSet 𝑦) |
12 | 10 | brsset 32599 | . . . . . 6 ⊢ (𝑧 SSet 𝑦 ↔ 𝑧 ⊆ 𝑦) |
13 | 11, 12 | bitri 267 | . . . . 5 ⊢ (𝑦◡ SSet 𝑧 ↔ 𝑧 ⊆ 𝑦) |
14 | 9, 13 | anbi12i 620 | . . . 4 ⊢ ((𝑦 SSet 𝑧 ∧ 𝑦◡ SSet 𝑧) ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦)) |
15 | 8, 14 | bitri 267 | . . 3 ⊢ (𝑦( SSet ∩ ◡ SSet )𝑧 ↔ (𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦)) |
16 | 5, 7, 15 | 3bitr4i 295 | . 2 ⊢ (𝑦 I 𝑧 ↔ 𝑦( SSet ∩ ◡ SSet )𝑧) |
17 | 1, 4, 16 | eqbrriv 5462 | 1 ⊢ I = ( SSet ∩ ◡ SSet ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 = wceq 1601 ∩ cin 3790 ⊆ wss 3791 class class class wbr 4886 I cid 5260 ◡ccnv 5354 Rel wrel 5360 SSet csset 32542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-eprel 5266 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-fo 6141 df-fv 6143 df-1st 7445 df-2nd 7446 df-txp 32564 df-sset 32566 |
This theorem is referenced by: (None) |
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