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Theorem refimssco 36932
 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 4587 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝑥𝐴𝑧𝑥𝐴𝑥))
2 breq1 4586 . . . . . . . . . . 11 (𝑧 = 𝑥 → (𝑧𝐴𝑦𝑥𝐴𝑦))
31, 2anbi12d 743 . . . . . . . . . 10 (𝑧 = 𝑥 → ((𝑥𝐴𝑧𝑧𝐴𝑦) ↔ (𝑥𝐴𝑥𝑥𝐴𝑦)))
43biimprd 237 . . . . . . . . 9 (𝑧 = 𝑥 → ((𝑥𝐴𝑥𝑥𝐴𝑦) → (𝑥𝐴𝑧𝑧𝐴𝑦)))
54spimev 2247 . . . . . . . 8 ((𝑥𝐴𝑥𝑥𝐴𝑦) → ∃𝑧(𝑥𝐴𝑧𝑧𝐴𝑦))
65ex 449 . . . . . . 7 (𝑥𝐴𝑥 → (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐴𝑧𝑧𝐴𝑦)))
76adantr 480 . . . . . 6 ((𝑥𝐴𝑥𝑦𝐴𝑦) → (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐴𝑧𝑧𝐴𝑦)))
87com12 32 . . . . 5 (𝑥𝐴𝑦 → ((𝑥𝐴𝑥𝑦𝐴𝑦) → ∃𝑧(𝑥𝐴𝑧𝑧𝐴𝑦)))
98a2i 14 . . . 4 ((𝑥𝐴𝑦 → (𝑥𝐴𝑥𝑦𝐴𝑦)) → (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐴𝑧𝑧𝐴𝑦)))
10 19.37v 1897 . . . 4 (∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧𝑧𝐴𝑦)) ↔ (𝑥𝐴𝑦 → ∃𝑧(𝑥𝐴𝑧𝑧𝐴𝑦)))
119, 10sylibr 223 . . 3 ((𝑥𝐴𝑦 → (𝑥𝐴𝑥𝑦𝐴𝑦)) → ∃𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧𝑧𝐴𝑦)))
12112alimi 1731 . 2 (∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥𝑦𝐴𝑦)) → ∀𝑥𝑦𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧𝑧𝐴𝑦)))
13 reflexg 36930 . 2 (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 ↔ ∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥𝑦𝐴𝑦)))
14 cnvssco 36931 . 2 (𝐴(𝐴𝐴) ↔ ∀𝑥𝑦𝑧(𝑥𝐴𝑦 → (𝑥𝐴𝑧𝑧𝐴𝑦)))
1512, 13, 143imtr4i 280 1 (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴𝐴(𝐴𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473  ∃wex 1695   ∪ cun 3538   ⊆ wss 3540   class class class wbr 4583   I cid 4948  ◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040   ∘ ccom 5042 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050 This theorem is referenced by: (None)
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