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Mirrors > Home > ILE Home > Th. List > rescnvcnv | GIF version |
Description: The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
rescnvcnv | ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvcnv2 5062 | . . 3 ⊢ ◡◡𝐴 = (𝐴 ↾ V) | |
2 | 1 | reseq1i 4885 | . 2 ⊢ (◡◡𝐴 ↾ 𝐵) = ((𝐴 ↾ V) ↾ 𝐵) |
3 | resres 4901 | . 2 ⊢ ((𝐴 ↾ V) ↾ 𝐵) = (𝐴 ↾ (V ∩ 𝐵)) | |
4 | ssv 3169 | . . . 4 ⊢ 𝐵 ⊆ V | |
5 | sseqin2 3346 | . . . 4 ⊢ (𝐵 ⊆ V ↔ (V ∩ 𝐵) = 𝐵) | |
6 | 4, 5 | mpbi 144 | . . 3 ⊢ (V ∩ 𝐵) = 𝐵 |
7 | 6 | reseq2i 4886 | . 2 ⊢ (𝐴 ↾ (V ∩ 𝐵)) = (𝐴 ↾ 𝐵) |
8 | 2, 3, 7 | 3eqtri 2195 | 1 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 Vcvv 2730 ∩ cin 3120 ⊆ wss 3121 ◡ccnv 4608 ↾ cres 4611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-xp 4615 df-rel 4616 df-cnv 4617 df-res 4621 |
This theorem is referenced by: cnvcnvres 5072 imacnvcnv 5073 resdm2 5099 resdmres 5100 coires1 5126 cocnvres 5133 f1oresrab 5659 |
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