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Theorem refimssco 41821
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 5107 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝑥𝐴𝑧𝑥𝐴𝑥))
2 breq1 5106 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝑧𝐴𝑦𝑥𝐴𝑦))
31, 2anbi12d 631 . . . . . . . . . 10 (𝑧 = 𝑥 → ((𝑥𝐴𝑧𝑧𝐴𝑦) ↔ (𝑥𝐴𝑥𝑥𝐴𝑦)))
43biimprd 247 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝑥𝐴𝑥𝑥𝐴𝑦) → (𝑥𝐴𝑧𝑧𝐴𝑦)))
54spimevw 1998 . . . . . . . 8 ((𝑥𝐴𝑥𝑥𝐴𝑦) → ∃𝑧(𝑥𝐴𝑧𝑧𝐴𝑦))
65ex 413 . . . . . . 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 1995 . . . 4 (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧𝑧𝐴𝑦)) ↔ (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐴𝑧𝑧𝐴𝑦)))
119, 10sylibr 233 . . 3 ((𝑥𝐴𝑦 → (𝑥𝐴𝑥𝑦𝐴𝑦)) → ∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧𝑧𝐴𝑦)))
12112alimi 1814 . 2 (∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥𝑦𝐴𝑦)) → ∀𝑥𝑦𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧𝑧𝐴𝑦)))
13 reflexg 41819 . 2 (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 ↔ ∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥𝑦𝐴𝑦)))
14 cnvssco 41820 . 2 (𝐴(𝐴𝐴) ↔ ∀𝑥𝑦𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧𝑧𝐴𝑦)))
1512, 13, 143imtr4i 291 1 (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴𝐴(𝐴𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1539  wex 1781  cun 3906  wss 3908   class class class wbr 5103   I cid 5528  ccnv 5630  dom cdm 5631  ran crn 5632  cres 5633  ccom 5635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643
This theorem is referenced by: (None)
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