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Theorem relresfld 4928
Description: Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)
Assertion
Ref Expression
relresfld (Rel 𝑅 → (𝑅 𝑅) = 𝑅)

Proof of Theorem relresfld
StepHypRef Expression
1 relfld 4927 . . . 4 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
21reseq2d 4683 . . 3 (Rel 𝑅 → (𝑅 𝑅) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)))
3 resundi 4696 . . 3 (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅))
4 eqtr 2102 . . . 4 (((𝑅 𝑅) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ∧ (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅))) → (𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)))
5 resss 4706 . . . . 5 (𝑅 ↾ ran 𝑅) ⊆ 𝑅
6 resdm 4720 . . . . 5 (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
7 ssequn2 3162 . . . . . 6 ((𝑅 ↾ ran 𝑅) ⊆ 𝑅 ↔ (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅)
8 uneq1 3136 . . . . . . . . 9 ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)))
98eqeq2d 2096 . . . . . . . 8 ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) ↔ (𝑅 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅))))
10 eqtr 2102 . . . . . . . . 9 (((𝑅 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)) ∧ (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅) → (𝑅 𝑅) = 𝑅)
1110ex 113 . . . . . . . 8 ((𝑅 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)) → ((𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅 → (𝑅 𝑅) = 𝑅))
129, 11syl6bi 161 . . . . . . 7 ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) → ((𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅 → (𝑅 𝑅) = 𝑅)))
1312com3r 78 . . . . . 6 ((𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅 → ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) → (𝑅 𝑅) = 𝑅)))
147, 13sylbi 119 . . . . 5 ((𝑅 ↾ ran 𝑅) ⊆ 𝑅 → ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) → (𝑅 𝑅) = 𝑅)))
155, 6, 14mpsyl 64 . . . 4 (Rel 𝑅 → ((𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) → (𝑅 𝑅) = 𝑅))
164, 15syl5com 29 . . 3 (((𝑅 𝑅) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ∧ (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅))) → (Rel 𝑅 → (𝑅 𝑅) = 𝑅))
172, 3, 16sylancl 404 . 2 (Rel 𝑅 → (Rel 𝑅 → (𝑅 𝑅) = 𝑅))
1817pm2.43i 48 1 (Rel 𝑅 → (𝑅 𝑅) = 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1287  cun 2986  wss 2988   cuni 3638  dom cdm 4413  ran crn 4414  cres 4415  Rel wrel 4418
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-xp 4419  df-rel 4420  df-cnv 4421  df-dm 4423  df-rn 4424  df-res 4425
This theorem is referenced by: (None)
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