Theorem cossssid5 38427

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 38426	. 2 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥)
2		ineccnvmo2 38316	. 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 846  ∀wal 1535   = wceq 1537  ∃*wmo 2541  ∀wral 3067   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352   class class class wbr 5166   I cid 5592  ^◡ccnv 5699  ran crn 5701  [cec 8761   ≀ ccoss 38135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rmo 3388  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ec 8765  df-coss 38367

This theorem is referenced by:  cosselcnvrefrels5  38497  dffunALTV5  38647