Theorem reflexg 38228

Description: Two ways of saying a relation is reflexive over its domain and range. (Contributed by RP, 4-Aug-2020.)

Assertion

Ref	Expression
reflexg	⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦)))

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem reflexg

Step	Hyp	Ref	Expression
1		undmrnresiss 38227	1 ⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1521   ∪ cun 3605   ⊆ wss 3607   class class class wbr 4685   I cid 5052  dom cdm 5143  ran crn 5144   ↾ cres 5145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155

This theorem is referenced by:  refimssco  38230