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Theorem refimssco 37732
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 4648 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝑥𝐴𝑧𝑥𝐴𝑥))
2 breq1 4647 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝑧𝐴𝑦𝑥𝐴𝑦))
31, 2anbi12d 746 . . . . . . . . . 10 (𝑧 = 𝑥 → ((𝑥𝐴𝑧𝑧𝐴𝑦) ↔ (𝑥𝐴𝑥𝑥𝐴𝑦)))
43biimprd 238 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝑥𝐴𝑥𝑥𝐴𝑦) → (𝑥𝐴𝑧𝑧𝐴𝑦)))
54spimev 2257 . . . . . . . 8 ((𝑥𝐴𝑥𝑥𝐴𝑦) → ∃𝑧(𝑥𝐴𝑧𝑧𝐴𝑦))
65ex 450 . . . . . . 7 (𝑥𝐴𝑥 → (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐴𝑧𝑧𝐴𝑦)))
76adantr 481 . . . . . 6 ((𝑥𝐴𝑥𝑦𝐴𝑦) → (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐴𝑧𝑧𝐴𝑦)))
87com12 32 . . . . 5 (𝑥𝐴𝑦 → ((𝑥𝐴𝑥𝑦𝐴𝑦) → ∃𝑧(𝑥𝐴𝑧𝑧𝐴𝑦)))
98a2i 14 . . . 4 ((𝑥𝐴𝑦 → (𝑥𝐴𝑥𝑦𝐴𝑦)) → (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐴𝑧𝑧𝐴𝑦)))
10 19.37v 1908 . . . 4 (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧𝑧𝐴𝑦)) ↔ (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐴𝑧𝑧𝐴𝑦)))
119, 10sylibr 224 . . 3 ((𝑥𝐴𝑦 → (𝑥𝐴𝑥𝑦𝐴𝑦)) → ∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧𝑧𝐴𝑦)))
12112alimi 1738 . 2 (∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥𝑦𝐴𝑦)) → ∀𝑥𝑦𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧𝑧𝐴𝑦)))
13 reflexg 37730 . 2 (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 ↔ ∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥𝑦𝐴𝑦)))
14 cnvssco 37731 . 2 (𝐴(𝐴𝐴) ↔ ∀𝑥𝑦𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧𝑧𝐴𝑦)))
1512, 13, 143imtr4i 281 1 (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴𝐴(𝐴𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1479  wex 1702  cun 3565  wss 3567   class class class wbr 4644   I cid 5013  ccnv 5103  dom cdm 5104  ran crn 5105  cres 5106  ccom 5108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116
This theorem is referenced by: (None)
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