Theorem cosscnvssid3 38577

Description: Equivalent expressions for the class of cosets by the converse of 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 28-Jul-2021.)

Assertion

Ref	Expression
cosscnvssid3	⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣))

Distinct variable group:   𝑢,𝑅,𝑣,𝑥

Proof of Theorem cosscnvssid3

Step	Hyp	Ref	Expression
1		cossssid3 38570	. 2 ⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑥∀𝑢∀𝑣((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣))
2		alrot3 2163	. 2 ⊢ (∀𝑥∀𝑢∀𝑣((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣) ↔ ∀𝑢∀𝑣∀𝑥((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣))
3		brcnvg 5818	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥))
4	3	el2v 3443	. . . . 5 ⊢ (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥)
5		brcnvg 5818	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑣 ∈ V) → (𝑥◡𝑅𝑣 ↔ 𝑣𝑅𝑥))
6	5	el2v 3443	. . . . 5 ⊢ (𝑥◡𝑅𝑣 ↔ 𝑣𝑅𝑥)
7	4, 6	anbi12i 628	. . . 4 ⊢ ((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) ↔ (𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥))
8	7	imbi1i 349	. . 3 ⊢ (((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣) ↔ ((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣))
9	8	3albii 1822	. 2 ⊢ (∀𝑢∀𝑣∀𝑥((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣) ↔ ∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣))
10	1, 2, 9	3bitri 297	1 ⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539  Vcvv 3436   ⊆ wss 3897   class class class wbr 5089   I cid 5508  ^◡ccnv 5613   ≀ 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-rab 3396  df-v 3438  df-dif 3900  df-un 3902  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-cnv 5622  df-coss 38512

This theorem is referenced by:  dfdisjs3  38807  dfdisjALTV3  38812  eldisjs3  38821