Theorem cosscnvssid3 36521

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 36514	. 2 ⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑥∀𝑢∀𝑣((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣))
2		alrot3 2159	. 2 ⊢ (∀𝑥∀𝑢∀𝑣((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣) ↔ ∀𝑢∀𝑣∀𝑥((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣))
3		brcnvg 5777	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥))
4	3	el2v 3430	. . . . 5 ⊢ (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥)
5		brcnvg 5777	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑣 ∈ V) → (𝑥◡𝑅𝑣 ↔ 𝑣𝑅𝑥))
6	5	el2v 3430	. . . . 5 ⊢ (𝑥◡𝑅𝑣 ↔ 𝑣𝑅𝑥)
7	4, 6	anbi12i 626	. . . 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 36314	. 2 ⊢ (∀𝑢∀𝑣∀𝑥((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣) ↔ ∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣))
10	1, 2, 9	3bitri 296	1 ⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537  Vcvv 3422   ⊆ wss 3883   class class class wbr 5070   I cid 5479  ^◡ccnv 5579   ≀ ccoss 36260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5480  df-cnv 5588  df-coss 36464

This theorem is referenced by:  dfdisjs3  36748  dfdisjALTV3  36753  eldisjs3  36762