Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dfrel4v | Structured version Visualization version GIF version |
Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6726 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) |
Ref | Expression |
---|---|
dfrel4v | ⊢ (Rel 𝑅 ↔ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrel2 6048 | . 2 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
2 | eqcom 2830 | . 2 ⊢ (◡◡𝑅 = 𝑅 ↔ 𝑅 = ◡◡𝑅) | |
3 | cnvcnv3 6047 | . . 3 ⊢ ◡◡𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦} | |
4 | 3 | eqeq2i 2836 | . 2 ⊢ (𝑅 = ◡◡𝑅 ↔ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦}) |
5 | 1, 2, 4 | 3bitri 299 | 1 ⊢ (Rel 𝑅 ↔ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝑥𝑅𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 class class class wbr 5068 {copab 5130 ◡ccnv 5556 Rel wrel 5562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 |
This theorem is referenced by: dfrel4 6050 dffn5 6726 fsplit 7814 fsplitOLD 7815 pwsle 16767 tgphaus 22727 fneer 33703 inxp2 35621 dfxrn2 35630 1cosscnvxrn 35717 dfafn5a 43366 sprsymrelfo 43666 |
Copyright terms: Public domain | W3C validator |