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Theorem relresfld 5904
 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 5903 . . . 4 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
21reseq2d 5630 . . 3 (Rel 𝑅 → (𝑅 𝑅) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)))
3 resundi 5648 . . 3 (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅))
4 eqtr 2847 . . . 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 5659 . . . . 5 (𝑅 ↾ ran 𝑅) ⊆ 𝑅
6 resdm 5679 . . . . 5 (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
7 ssequn2 4014 . . . . . 6 ((𝑅 ↾ ran 𝑅) ⊆ 𝑅 ↔ (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅)
8 uneq1 3988 . . . . . . . . 9 ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)))
98eqeq2d 2836 . . . . . . . 8 ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) ↔ (𝑅 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅))))
10 eqtr 2847 . . . . . . . . 9 (((𝑅 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)) ∧ (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅) → (𝑅 𝑅) = 𝑅)
1110ex 403 . . . . . . . 8 ((𝑅 𝑅) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)) → ((𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅 → (𝑅 𝑅) = 𝑅))
129, 11syl6bi 245 . . . . . . 7 ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) → ((𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅 → (𝑅 𝑅) = 𝑅)))
1312com3r 87 . . . . . 6 ((𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅 → ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) → (𝑅 𝑅) = 𝑅)))
147, 13sylbi 209 . . . . 5 ((𝑅 ↾ ran 𝑅) ⊆ 𝑅 → ((𝑅 ↾ dom 𝑅) = 𝑅 → ((𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) → (𝑅 𝑅) = 𝑅)))
155, 6, 14mpsyl 68 . . . 4 (Rel 𝑅 → ((𝑅 𝑅) = ((𝑅 ↾ dom 𝑅) ∪ (𝑅 ↾ ran 𝑅)) → (𝑅 𝑅) = 𝑅))
164, 15syl5com 31 . . 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 582 . 2 (Rel 𝑅 → (Rel 𝑅 → (𝑅 𝑅) = 𝑅))
1817pm2.43i 52 1 (Rel 𝑅 → (𝑅 𝑅) = 𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∪ cun 3797   ⊆ wss 3799  ∪ cuni 4659  dom cdm 5343  ran crn 5344   ↾ cres 5345  Rel wrel 5348 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-xp 5349  df-rel 5350  df-cnv 5351  df-dm 5353  df-rn 5354  df-res 5355 This theorem is referenced by:  relcoi1  5906
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