Theorem dfcnvrefrel3 35773

Description: Alternate definition of the converse reflexive relation predicate. A relation is converse reflexive iff: for all elements on its domain and range, if for an element of its domain and for an element of its range there is the relation between them, then the two elements are the same, cf. the comment of dfrefrel3 35760. (Contributed by Peter Mazsa, 25-Jul-2021.)

Assertion

Ref	Expression
dfcnvrefrel3	⊢ ( CnvRefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦 → 𝑥 = 𝑦) ∧ Rel 𝑅))

Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem dfcnvrefrel3

Step	Hyp	Ref	Expression
1		df-cnvrefrel 35769	. 2 ⊢ ( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
2		inxpssidinxp 35577	. . 3 ⊢ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ↔ ∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦 → 𝑥 = 𝑦))
3	2	anbi1i 625	. 2 ⊢ (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅) ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦 → 𝑥 = 𝑦) ∧ Rel 𝑅))
4	1, 3	bitri 277	1 ⊢ ( CnvRefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦 → 𝑥 = 𝑦) ∧ Rel 𝑅))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wral 3141   ∩ cin 3938   ⊆ wss 3939   class class class wbr 5069   I cid 5462   × cxp 5556  dom cdm 5558  ran crn 5559  Rel wrel 5563   CnvRefRel wcnvrefrel 35466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-id 5463  df-xp 5564  df-rel 5565  df-cnvrefrel 35769

This theorem is referenced by: (None)