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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 6213 | . . 3 ⊢ ◡◡𝐴 = (𝐴 ↾ V) | |
| 2 | 1 | reseq1i 5993 | . 2 ⊢ (◡◡𝐴 ↾ 𝐵) = ((𝐴 ↾ V) ↾ 𝐵) | 
| 3 | resres 6010 | . 2 ⊢ ((𝐴 ↾ V) ↾ 𝐵) = (𝐴 ↾ (V ∩ 𝐵)) | |
| 4 | ssv 4008 | . . . 4 ⊢ 𝐵 ⊆ V | |
| 5 | sseqin2 4223 | . . . 4 ⊢ (𝐵 ⊆ V ↔ (V ∩ 𝐵) = 𝐵) | |
| 6 | 4, 5 | mpbi 230 | . . 3 ⊢ (V ∩ 𝐵) = 𝐵 | 
| 7 | 6 | reseq2i 5994 | . 2 ⊢ (𝐴 ↾ (V ∩ 𝐵)) = (𝐴 ↾ 𝐵) | 
| 8 | 2, 3, 7 | 3eqtri 2769 | 1 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ◡ccnv 5684 ↾ cres 5687 | 
| 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-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 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-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-res 5697 | 
| This theorem is referenced by: cnvcnvres 6225 imacnvcnv 6226 resdm2 6251 resdmres 6252 coires1 6284 f1oresrab 7147 | 
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