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Theorem refimssco 37433
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 4627 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝑥𝐴𝑧𝑥𝐴𝑥))
2 breq1 4626 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝑧𝐴𝑦𝑥𝐴𝑦))
31, 2anbi12d 746 . . . . . . . . . 10 (𝑧 = 𝑥 → ((𝑥𝐴𝑧𝑧𝐴𝑦) ↔ (𝑥𝐴𝑥𝑥𝐴𝑦)))
43biimprd 238 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝑥𝐴𝑥𝑥𝐴𝑦) → (𝑥𝐴𝑧𝑧𝐴𝑦)))
54spimev 2258 . . . . . . . 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 1907 . . . 4 (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧𝑧𝐴𝑦)) ↔ (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐴𝑧𝑧𝐴𝑦)))
119, 10sylibr 224 . . 3 ((𝑥𝐴𝑦 → (𝑥𝐴𝑥𝑦𝐴𝑦)) → ∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧𝑧𝐴𝑦)))
12112alimi 1737 . 2 (∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥𝑦𝐴𝑦)) → ∀𝑥𝑦𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧𝑧𝐴𝑦)))
13 reflexg 37431 . 2 (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 ↔ ∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥𝑦𝐴𝑦)))
14 cnvssco 37432 . 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 1478  wex 1701  cun 3558  wss 3560   class class class wbr 4623   I cid 4994  ccnv 5083  dom cdm 5084  ran crn 5085  cres 5086  ccom 5088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096
This theorem is referenced by: (None)
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