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Mirrors > Home > MPE Home > Th. List > Mathboxes > cossssid5 | Structured version Visualization version GIF version |
Description: Equivalent expressions for the class of cosets by 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 5-Sep-2021.) |
Ref | Expression |
---|---|
cossssid5 | ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑥 ∈ ran 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]◡𝑅 ∩ [𝑦]◡𝑅) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cossssid4 35870 | . 2 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥) | |
2 | ineccnvmo2 35774 | . 2 ⊢ (∀𝑥 ∈ ran 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]◡𝑅 ∩ [𝑦]◡𝑅) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥) | |
3 | 1, 2 | bitr4i 281 | 1 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑥 ∈ ran 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]◡𝑅 ∩ [𝑦]◡𝑅) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∨ wo 844 ∀wal 1536 = wceq 1538 ∃*wmo 2596 ∀wral 3106 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 class class class wbr 5030 I cid 5424 ◡ccnv 5518 ran crn 5520 [cec 8270 ≀ ccoss 35613 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ec 8274 df-coss 35819 |
This theorem is referenced by: cosselcnvrefrels5 35937 dffunALTV5 36084 |
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