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Theorem relresfld 4790
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 4789 . . . 4 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
21reseq2d 4555 . . 3 (Rel 𝑅 → (𝑅 𝑅) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)))
3 resundi 4568 . . 3 (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅))
4 eqtr 2054 . . . 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 4578 . . . . 5 (𝑅 ↾ ran 𝑅) ⊆ 𝑅
6 resdm 4592 . . . . 5 (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
7 ssequn2 3110 . . . . . 6 ((𝑅 ↾ ran 𝑅) ⊆ 𝑅 ↔ (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅)
8 uneq1 3084 . . . . . . . . 9 ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)))
98eqeq2d 2048 . . . . . . . 8 ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) ↔ (𝑅 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅))))
10 eqtr 2054 . . . . . . . . 9 (((𝑅 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)) (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅) → (𝑅 𝑅) = 𝑅)
1110ex 108 . . . . . . . 8 ((𝑅 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)) → ((𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅 → (𝑅 𝑅) = 𝑅))
129, 11syl6bi 152 . . . . . . 7 ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) → ((𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅 → (𝑅 𝑅) = 𝑅)))
1312com3r 73 . . . . . 6 ((𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅 → ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) → (𝑅 𝑅) = 𝑅)))
147, 13sylbi 114 . . . . 5 ((𝑅 ↾ ran 𝑅) ⊆ 𝑅 → ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) → (𝑅 𝑅) = 𝑅)))
155, 6, 14mpsyl 59 . . . 4 (Rel 𝑅 → ((𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) → (𝑅 𝑅) = 𝑅))
164, 15syl5com 26 . . 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 392 . 2 (Rel 𝑅 → (Rel 𝑅 → (𝑅 𝑅) = 𝑅))
1817pm2.43i 43 1 (Rel 𝑅 → (𝑅 𝑅) = 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  cun 2909  wss 2911   cuni 3571  dom cdm 4288  ran crn 4289  cres 4290  Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300
This theorem is referenced by: (None)
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