Theorem cossssid5 38467

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.)

Assertion

Ref	Expression
cossssid5	⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑥 ∈ ran 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]◡𝑅 ∩ [𝑦]◡𝑅) = ∅))

Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem cossssid5

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cossssid4 38466	. 2 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥)
2		ineccnvmo2 38347	. 2 ⊢ (∀𝑥 ∈ ran 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]◡𝑅 ∩ [𝑦]◡𝑅) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥)
3	1, 2	bitr4i 278	1 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑥 ∈ ran 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]◡𝑅 ∩ [𝑦]◡𝑅) = ∅))

Syntax hints:   ↔ wb 206   ∨ wo 847  ∀wal 1538   = wceq 1540  ∃*wmo 2531  ∀wral 3044   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286   class class class wbr 5095   I cid 5517  ^◡ccnv 5622  ran crn 5624  [cec 8630   ≀ ccoss 38174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rmo 3345  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ec 8634  df-coss 38407

This theorem is referenced by:  cosselcnvrefrels5  38537  dffunALTV5  38688