Theorem cossssid2 35189

Description: Equivalent expressions for the class of cosets by 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 10-Mar-2019.)

Assertion

Ref	Expression
cossssid2	⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑥∀𝑦(∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))

Distinct variable group:   𝑢,𝑅,𝑥,𝑦

Proof of Theorem cossssid2

Step	Hyp	Ref	Expression
1		df-id 5340	. . 3 ⊢ I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
2	1	sseq2i 3912	. 2 ⊢ ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦})
3		df-coss 35140	. . 3 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
4	3	sseq1i 3911	. 2 ⊢ ( ≀ 𝑅 ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} ↔ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦})
5		ssopab2b 5316	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} ↔ ∀𝑥∀𝑦(∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))
6	2, 4, 5	3bitri 298	1 ⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑥∀𝑦(∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1518  ∃wex 1759   ⊆ wss 3854   class class class wbr 4956  {copab 5018   I cid 5339   ≀ 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-opab 5019  df-id 5340  df-coss 35140

This theorem is referenced by:  cossssid3  35190