Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  refimssco Structured version   Visualization version   GIF version

Theorem refimssco 38696
Description: Reflexive relations are subsets of their self-composition. (Contributed by RP, 4-Aug-2020.)
Assertion
Ref Expression
refimssco (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴𝐴(𝐴𝐴))

Proof of Theorem refimssco
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 4847 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝑥𝐴𝑧𝑥𝐴𝑥))
2 breq1 4846 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝑧𝐴𝑦𝑥𝐴𝑦))
31, 2anbi12d 625 . . . . . . . . . 10 (𝑧 = 𝑥 → ((𝑥𝐴𝑧𝑧𝐴𝑦) ↔ (𝑥𝐴𝑥𝑥𝐴𝑦)))
43biimprd 240 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝑥𝐴𝑥𝑥𝐴𝑦) → (𝑥𝐴𝑧𝑧𝐴𝑦)))
54spimev 2399 . . . . . . . 8 ((𝑥𝐴𝑥𝑥𝐴𝑦) → ∃𝑧(𝑥𝐴𝑧𝑧𝐴𝑦))
65ex 402 . . . . . . 7 (𝑥𝐴𝑥 → (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐴𝑧𝑧𝐴𝑦)))
76adantr 473 . . . . . 6 ((𝑥𝐴𝑥𝑦𝐴𝑦) → (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐴𝑧𝑧𝐴𝑦)))
87com12 32 . . . . 5 (𝑥𝐴𝑦 → ((𝑥𝐴𝑥𝑦𝐴𝑦) → ∃𝑧(𝑥𝐴𝑧𝑧𝐴𝑦)))
98a2i 14 . . . 4 ((𝑥𝐴𝑦 → (𝑥𝐴𝑥𝑦𝐴𝑦)) → (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐴𝑧𝑧𝐴𝑦)))
10 19.37v 2092 . . . 4 (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧𝑧𝐴𝑦)) ↔ (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐴𝑧𝑧𝐴𝑦)))
119, 10sylibr 226 . . 3 ((𝑥𝐴𝑦 → (𝑥𝐴𝑥𝑦𝐴𝑦)) → ∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧𝑧𝐴𝑦)))
12112alimi 1908 . 2 (∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥𝑦𝐴𝑦)) → ∀𝑥𝑦𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧𝑧𝐴𝑦)))
13 reflexg 38694 . 2 (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 ↔ ∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥𝑦𝐴𝑦)))
14 cnvssco 38695 . 2 (𝐴(𝐴𝐴) ↔ ∀𝑥𝑦𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧𝑧𝐴𝑦)))
1512, 13, 143imtr4i 284 1 (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴𝐴(𝐴𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wal 1651  wex 1875  cun 3767  wss 3769   class class class wbr 4843   I cid 5219  ccnv 5311  dom cdm 5312  ran crn 5313  cres 5314  ccom 5316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator