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Theorem sofld 6015
Description: The base set of a nonempty strict order is the same as the field of the relation. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
sofld ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → 𝐴 = (dom 𝑅 ∪ ran 𝑅))

Proof of Theorem sofld
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 5541 . . . . . . . . 9 Rel (𝐴 × 𝐴)
2 relss 5624 . . . . . . . . 9 (𝑅 ⊆ (𝐴 × 𝐴) → (Rel (𝐴 × 𝐴) → Rel 𝑅))
31, 2mpi 20 . . . . . . . 8 (𝑅 ⊆ (𝐴 × 𝐴) → Rel 𝑅)
43ad2antlr 726 . . . . . . 7 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → Rel 𝑅)
5 df-br 5034 . . . . . . . . . 10 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
6 ssun1 4102 . . . . . . . . . . . . 13 𝐴 ⊆ (𝐴 ∪ {𝑥})
7 undif1 4385 . . . . . . . . . . . . 13 ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥})
86, 7sseqtrri 3955 . . . . . . . . . . . 12 𝐴 ⊆ ((𝐴 ∖ {𝑥}) ∪ {𝑥})
9 simpll 766 . . . . . . . . . . . . . 14 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑅 Or 𝐴)
10 dmss 5739 . . . . . . . . . . . . . . . . 17 (𝑅 ⊆ (𝐴 × 𝐴) → dom 𝑅 ⊆ dom (𝐴 × 𝐴))
11 dmxpid 5768 . . . . . . . . . . . . . . . . 17 dom (𝐴 × 𝐴) = 𝐴
1210, 11sseqtrdi 3968 . . . . . . . . . . . . . . . 16 (𝑅 ⊆ (𝐴 × 𝐴) → dom 𝑅𝐴)
1312ad2antlr 726 . . . . . . . . . . . . . . 15 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → dom 𝑅𝐴)
143ad2antlr 726 . . . . . . . . . . . . . . . 16 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → Rel 𝑅)
15 releldm 5782 . . . . . . . . . . . . . . . 16 ((Rel 𝑅𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
1614, 15sylancom 591 . . . . . . . . . . . . . . 15 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
1713, 16sseldd 3919 . . . . . . . . . . . . . 14 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥𝐴)
18 sossfld 6014 . . . . . . . . . . . . . 14 ((𝑅 Or 𝐴𝑥𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
199, 17, 18syl2anc 587 . . . . . . . . . . . . 13 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
20 ssun1 4102 . . . . . . . . . . . . . . 15 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2120, 16sseldi 3916 . . . . . . . . . . . . . 14 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅))
2221snssd 4705 . . . . . . . . . . . . 13 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → {𝑥} ⊆ (dom 𝑅 ∪ ran 𝑅))
2319, 22unssd 4116 . . . . . . . . . . . 12 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
248, 23sstrid 3929 . . . . . . . . . . 11 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅))
2524ex 416 . . . . . . . . . 10 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) → (𝑥𝑅𝑦𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
265, 25syl5bir 246 . . . . . . . . 9 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) → (⟨𝑥, 𝑦⟩ ∈ 𝑅𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
2726con3dimp 412 . . . . . . . 8 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → ¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
2827pm2.21d 121 . . . . . . 7 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ ∅))
294, 28relssdv 5629 . . . . . 6 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → 𝑅 ⊆ ∅)
30 ss0 4309 . . . . . 6 (𝑅 ⊆ ∅ → 𝑅 = ∅)
3129, 30syl 17 . . . . 5 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → 𝑅 = ∅)
3231ex 416 . . . 4 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) → (¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅) → 𝑅 = ∅))
3332necon1ad 3007 . . 3 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) → (𝑅 ≠ ∅ → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
34333impia 1114 . 2 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅))
35 rnss 5777 . . . . 5 (𝑅 ⊆ (𝐴 × 𝐴) → ran 𝑅 ⊆ ran (𝐴 × 𝐴))
36 rnxpid 6001 . . . . 5 ran (𝐴 × 𝐴) = 𝐴
3735, 36sseqtrdi 3968 . . . 4 (𝑅 ⊆ (𝐴 × 𝐴) → ran 𝑅𝐴)
3812, 37unssd 4116 . . 3 (𝑅 ⊆ (𝐴 × 𝐴) → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝐴)
39383ad2ant2 1131 . 2 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝐴)
4034, 39eqssd 3935 1 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → 𝐴 = (dom 𝑅 ∪ ran 𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2112  wne 2990  cdif 3881  cun 3882  wss 3884  c0 4246  {csn 4528  cop 4534   class class class wbr 5033   Or wor 5441   × cxp 5521  dom cdm 5523  ran crn 5524  Rel wrel 5528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-po 5442  df-so 5443  df-xp 5529  df-rel 5530  df-cnv 5531  df-dm 5533  df-rn 5534
This theorem is referenced by: (None)
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