Theorem cosscnvssid3 35197

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 35190	. 2 ⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑥∀𝑢∀𝑣((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣))
2		alrot3 2127	. 2 ⊢ (∀𝑥∀𝑢∀𝑣((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣) ↔ ∀𝑢∀𝑣∀𝑥((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣))
3		brcnvg 5628	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥))
4	3	el2v 3439	. . . . 5 ⊢ (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥)
5		brcnvg 5628	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑣 ∈ V) → (𝑥◡𝑅𝑣 ↔ 𝑣𝑅𝑥))
6	5	el2v 3439	. . . . 5 ⊢ (𝑥◡𝑅𝑣 ↔ 𝑣𝑅𝑥)
7	4, 6	anbi12i 626	. . . 4 ⊢ ((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) ↔ (𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥))
8	7	imbi1i 351	. . 3 ⊢ (((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣) ↔ ((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣))
9	8	3albii 34989	. 2 ⊢ (∀𝑢∀𝑣∀𝑥((𝑥◡𝑅𝑢 ∧ 𝑥◡𝑅𝑣) → 𝑢 = 𝑣) ↔ ∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣))
10	1, 2, 9	3bitri 298	1 ⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1518  Vcvv 3432   ⊆ wss 3854   class class class wbr 4956   I cid 5339  ^◡ccnv 5434   ≀ ccoss 34933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pr 5214

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rab 3112  df-v 3434  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-br 4957  df-opab 5019  df-id 5340  df-cnv 5443  df-coss 35140

This theorem is referenced by:  dfdisjs3  35424  dfdisjALTV3  35429  eldisjs3  35438