![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6149 | . . 3 ⊢ ◡◡𝐴 = (𝐴 ↾ V) | |
2 | 1 | reseq1i 5937 | . 2 ⊢ (◡◡𝐴 ↾ 𝐵) = ((𝐴 ↾ V) ↾ 𝐵) |
3 | resres 5954 | . 2 ⊢ ((𝐴 ↾ V) ↾ 𝐵) = (𝐴 ↾ (V ∩ 𝐵)) | |
4 | ssv 3972 | . . . 4 ⊢ 𝐵 ⊆ V | |
5 | sseqin2 4179 | . . . 4 ⊢ (𝐵 ⊆ V ↔ (V ∩ 𝐵) = 𝐵) | |
6 | 4, 5 | mpbi 229 | . . 3 ⊢ (V ∩ 𝐵) = 𝐵 |
7 | 6 | reseq2i 5938 | . 2 ⊢ (𝐴 ↾ (V ∩ 𝐵)) = (𝐴 ↾ 𝐵) |
8 | 2, 3, 7 | 3eqtri 2765 | 1 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 Vcvv 3447 ∩ cin 3913 ⊆ wss 3914 ◡ccnv 5636 ↾ cres 5639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-xp 5643 df-rel 5644 df-cnv 5645 df-res 5649 |
This theorem is referenced by: cnvcnvres 6161 imacnvcnv 6162 resdm2 6187 resdmres 6188 coires1 6220 f1oresrab 7077 |
Copyright terms: Public domain | W3C validator |