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Mirrors > Home > MPE Home > Th. List > dfrel3 | Structured version Visualization version GIF version |
Description: Alternate definition of relation. (Contributed by NM, 14-May-2008.) |
Ref | Expression |
---|---|
dfrel3 | ⊢ (Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrel2 6177 | . 2 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
2 | cnvcnv2 6181 | . . 3 ⊢ ◡◡𝑅 = (𝑅 ↾ V) | |
3 | 2 | eqeq1i 2736 | . 2 ⊢ (◡◡𝑅 = 𝑅 ↔ (𝑅 ↾ V) = 𝑅) |
4 | 1, 3 | bitri 274 | 1 ⊢ (Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 Vcvv 3473 ◡ccnv 5668 ↾ cres 5671 Rel wrel 5674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 df-cnv 5677 df-res 5681 |
This theorem is referenced by: elid 6187 cocnvcnv2 6246 f1ovi 6859 ttrclco 9695 |
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