Theorem reflexg 44142

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 44141	1 ⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557   ∪ cun 3900   ⊆ wss 3902   class class class wbr 5097   I cid 5537  dom cdm 5643  ran crn 5644   ↾ cres 5645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-11 2190  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-dm 5653  df-rn 5654  df-res 5655

This theorem is referenced by:  refimssco  44144