Theorem cosscnvssid5 38579

Description: Equivalent expressions for the class of cosets by the converse of the relation 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 5-Sep-2021.)

Assertion

Ref	Expression
cosscnvssid5	⊢ (( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅) ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅))

Distinct variable group:   𝑢,𝑅,𝑣

Proof of Theorem cosscnvssid5

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cosscnvssid4 38578	. . 3 ⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥)
2	1	anbi1i 624	. 2 ⊢ (( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅) ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ Rel 𝑅))
3		inecmo3 38392	. 2 ⊢ ((∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅) ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ Rel 𝑅))
4	2, 3	bitr4i 278	1 ⊢ (( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅) ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∨ wo 847  ∀wal 1539   = wceq 1541  ∃*wmo 2533  ∀wral 3047   ∩ cin 3896   ⊆ wss 3897  ∅c0 4280   class class class wbr 5089   I cid 5508  ^◡ccnv 5613  dom cdm 5614  Rel wrel 5619  [cec 8620   ≀ ccoss 38221

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rmo 3346  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ec 8624  df-coss 38512

This theorem is referenced by:  dfdisjs5  38809  dfdisjALTV5  38814  eldisjs5  38823