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Theorem sofld 5500
 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 5150 . . . . . . . . 9 Rel (𝐴 × 𝐴)
2 relss 5129 . . . . . . . . 9 (𝑅 ⊆ (𝐴 × 𝐴) → (Rel (𝐴 × 𝐴) → Rel 𝑅))
31, 2mpi 20 . . . . . . . 8 (𝑅 ⊆ (𝐴 × 𝐴) → Rel 𝑅)
43ad2antlr 759 . . . . . . 7 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → Rel 𝑅)
5 df-br 4584 . . . . . . . . . 10 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
6 ssun1 3738 . . . . . . . . . . . . 13 𝐴 ⊆ (𝐴 ∪ {𝑥})
7 undif1 3995 . . . . . . . . . . . . 13 ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥})
86, 7sseqtr4i 3601 . . . . . . . . . . . 12 𝐴 ⊆ ((𝐴 ∖ {𝑥}) ∪ {𝑥})
9 simpll 786 . . . . . . . . . . . . . 14 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑅 Or 𝐴)
10 dmss 5245 . . . . . . . . . . . . . . . . 17 (𝑅 ⊆ (𝐴 × 𝐴) → dom 𝑅 ⊆ dom (𝐴 × 𝐴))
11 dmxpid 5266 . . . . . . . . . . . . . . . . 17 dom (𝐴 × 𝐴) = 𝐴
1210, 11syl6sseq 3614 . . . . . . . . . . . . . . . 16 (𝑅 ⊆ (𝐴 × 𝐴) → dom 𝑅𝐴)
1312ad2antlr 759 . . . . . . . . . . . . . . 15 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → dom 𝑅𝐴)
143ad2antlr 759 . . . . . . . . . . . . . . . 16 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → Rel 𝑅)
15 releldm 5279 . . . . . . . . . . . . . . . 16 ((Rel 𝑅𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
1614, 15sylancom 698 . . . . . . . . . . . . . . 15 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
1713, 16sseldd 3569 . . . . . . . . . . . . . 14 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥𝐴)
18 sossfld 5499 . . . . . . . . . . . . . 14 ((𝑅 Or 𝐴𝑥𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
199, 17, 18syl2anc 691 . . . . . . . . . . . . 13 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
20 ssun1 3738 . . . . . . . . . . . . . . 15 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2120, 16sseldi 3566 . . . . . . . . . . . . . 14 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅))
2221snssd 4281 . . . . . . . . . . . . 13 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → {𝑥} ⊆ (dom 𝑅 ∪ ran 𝑅))
2319, 22unssd 3751 . . . . . . . . . . . 12 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
248, 23syl5ss 3579 . . . . . . . . . . 11 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅))
2524ex 449 . . . . . . . . . 10 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) → (𝑥𝑅𝑦𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
265, 25syl5bir 232 . . . . . . . . 9 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) → (⟨𝑥, 𝑦⟩ ∈ 𝑅𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
2726con3dimp 456 . . . . . . . 8 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → ¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
2827pm2.21d 117 . . . . . . 7 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ ∅))
294, 28relssdv 5135 . . . . . 6 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → 𝑅 ⊆ ∅)
30 ss0 3926 . . . . . 6 (𝑅 ⊆ ∅ → 𝑅 = ∅)
3129, 30syl 17 . . . . 5 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → 𝑅 = ∅)
3231ex 449 . . . 4 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) → (¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅) → 𝑅 = ∅))
3332necon1ad 2799 . . 3 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) → (𝑅 ≠ ∅ → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
34333impia 1253 . 2 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅))
35 rnss 5275 . . . . 5 (𝑅 ⊆ (𝐴 × 𝐴) → ran 𝑅 ⊆ ran (𝐴 × 𝐴))
36 rnxpid 5486 . . . . 5 ran (𝐴 × 𝐴) = 𝐴
3735, 36syl6sseq 3614 . . . 4 (𝑅 ⊆ (𝐴 × 𝐴) → ran 𝑅𝐴)
3812, 37unssd 3751 . . 3 (𝑅 ⊆ (𝐴 × 𝐴) → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝐴)
39383ad2ant2 1076 . 2 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝐴)
4034, 39eqssd 3585 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 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131   class class class wbr 4583   Or wor 4958   × cxp 5036  dom cdm 5038  ran crn 5039  Rel wrel 5043 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-3or 1032  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-ne 2782  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-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049 This theorem is referenced by: (None)
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