Theorem cosscnvssid3 36594

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 36587	. 2 ⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑥∀𝑢∀𝑣((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣))
2		alrot3 2157	. 2 ⊢ (∀𝑥∀𝑢∀𝑣((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣) ↔ ∀𝑢∀𝑣∀𝑥((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣))
3		brcnvg 5788	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥))
4	3	el2v 3440	. . . . 5 ⊢ (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥)
5		brcnvg 5788	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑣 ∈ V) → (𝑥◡𝑅𝑣 ↔ 𝑣𝑅𝑥))
6	5	el2v 3440	. . . . 5 ⊢ (𝑥◡𝑅𝑣 ↔ 𝑣𝑅𝑥)
7	4, 6	anbi12i 627	. . . 4 ⊢ ((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) ↔ (𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥))
8	7	imbi1i 350	. . 3 ⊢ (((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣) ↔ ((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣))
9	8	3albii 36387	. 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 205   ∧ wa 396  ∀wal 1537  Vcvv 3432   ⊆ wss 3887   class class class wbr 5074   I cid 5488  ^◡ccnv 5588   ≀ ccoss 36333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-cnv 5597  df-coss 36537

This theorem is referenced by:  dfdisjs3  36821  dfdisjALTV3  36826  eldisjs3  36835