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Mirrors > Home > MPE Home > Th. List > Mathboxes > relcnveq4 | Structured version Visualization version GIF version |
Description: Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 28-Apr-2019.) |
Ref | Expression |
---|---|
relcnveq4 | ⊢ (Rel 𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnveq 35611 | . 2 ⊢ (Rel 𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅)) | |
2 | relcnveq2 35612 | . 2 ⊢ (Rel 𝑅 → (◡𝑅 = 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥))) | |
3 | 1, 2 | bitrd 281 | 1 ⊢ (Rel 𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 = wceq 1537 ⊆ wss 3924 class class class wbr 5052 ◡ccnv 5540 Rel wrel 5546 |
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 2793 ax-sep 5189 ax-nul 5196 ax-pr 5316 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-br 5053 df-opab 5115 df-xp 5547 df-rel 5548 df-cnv 5549 |
This theorem is referenced by: dfsymrel5 35820 |
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