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Mirrors > Home > MPE Home > Th. List > rescnvcnv | Structured version Visualization version 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 6215 | . . 3 ⊢ ◡◡𝐴 = (𝐴 ↾ V) | |
2 | 1 | reseq1i 5996 | . 2 ⊢ (◡◡𝐴 ↾ 𝐵) = ((𝐴 ↾ V) ↾ 𝐵) |
3 | resres 6013 | . 2 ⊢ ((𝐴 ↾ V) ↾ 𝐵) = (𝐴 ↾ (V ∩ 𝐵)) | |
4 | ssv 4020 | . . . 4 ⊢ 𝐵 ⊆ V | |
5 | sseqin2 4231 | . . . 4 ⊢ (𝐵 ⊆ V ↔ (V ∩ 𝐵) = 𝐵) | |
6 | 4, 5 | mpbi 230 | . . 3 ⊢ (V ∩ 𝐵) = 𝐵 |
7 | 6 | reseq2i 5997 | . 2 ⊢ (𝐴 ↾ (V ∩ 𝐵)) = (𝐴 ↾ 𝐵) |
8 | 2, 3, 7 | 3eqtri 2767 | 1 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 ◡ccnv 5688 ↾ cres 5691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-res 5701 |
This theorem is referenced by: cnvcnvres 6227 imacnvcnv 6228 resdm2 6253 resdmres 6254 coires1 6286 f1oresrab 7147 |
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