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Theorem sofld 6139
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 5651 . . . . . . . . 9 Rel (𝐴 × 𝐴)
2 relss 5737 . . . . . . . . 9 (𝑅 ⊆ (𝐴 × 𝐴) → (Rel (𝐴 × 𝐴) → Rel 𝑅))
31, 2mpi 20 . . . . . . . 8 (𝑅 ⊆ (𝐴 × 𝐴) → Rel 𝑅)
43ad2antlr 725 . . . . . . 7 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → Rel 𝑅)
5 df-br 5106 . . . . . . . . . 10 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
6 ssun1 4132 . . . . . . . . . . . . 13 𝐴 ⊆ (𝐴 ∪ {𝑥})
7 undif1 4435 . . . . . . . . . . . . 13 ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥})
86, 7sseqtrri 3981 . . . . . . . . . . . 12 𝐴 ⊆ ((𝐴 ∖ {𝑥}) ∪ {𝑥})
9 simpll 765 . . . . . . . . . . . . . 14 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑅 Or 𝐴)
10 dmss 5858 . . . . . . . . . . . . . . . . 17 (𝑅 ⊆ (𝐴 × 𝐴) → dom 𝑅 ⊆ dom (𝐴 × 𝐴))
11 dmxpid 5885 . . . . . . . . . . . . . . . . 17 dom (𝐴 × 𝐴) = 𝐴
1210, 11sseqtrdi 3994 . . . . . . . . . . . . . . . 16 (𝑅 ⊆ (𝐴 × 𝐴) → dom 𝑅𝐴)
1312ad2antlr 725 . . . . . . . . . . . . . . 15 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → dom 𝑅𝐴)
143ad2antlr 725 . . . . . . . . . . . . . . . 16 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → Rel 𝑅)
15 releldm 5899 . . . . . . . . . . . . . . . 16 ((Rel 𝑅𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
1614, 15sylancom 588 . . . . . . . . . . . . . . 15 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
1713, 16sseldd 3945 . . . . . . . . . . . . . 14 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥𝐴)
18 sossfld 6138 . . . . . . . . . . . . . 14 ((𝑅 Or 𝐴𝑥𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
199, 17, 18syl2anc 584 . . . . . . . . . . . . 13 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
20 ssun1 4132 . . . . . . . . . . . . . . 15 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2120, 16sselid 3942 . . . . . . . . . . . . . 14 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅))
2221snssd 4769 . . . . . . . . . . . . 13 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → {𝑥} ⊆ (dom 𝑅 ∪ ran 𝑅))
2319, 22unssd 4146 . . . . . . . . . . . 12 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
248, 23sstrid 3955 . . . . . . . . . . 11 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅))
2524ex 413 . . . . . . . . . 10 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) → (𝑥𝑅𝑦𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
265, 25biimtrrid 242 . . . . . . . . 9 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) → (⟨𝑥, 𝑦⟩ ∈ 𝑅𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
2726con3dimp 409 . . . . . . . 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 5744 . . . . . 6 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → 𝑅 ⊆ ∅)
30 ss0 4358 . . . . . 6 (𝑅 ⊆ ∅ → 𝑅 = ∅)
3129, 30syl 17 . . . . 5 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → 𝑅 = ∅)
3231ex 413 . . . 4 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) → (¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅) → 𝑅 = ∅))
3332necon1ad 2960 . . 3 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) → (𝑅 ≠ ∅ → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
34333impia 1117 . 2 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅))
35 rnss 5894 . . . . 5 (𝑅 ⊆ (𝐴 × 𝐴) → ran 𝑅 ⊆ ran (𝐴 × 𝐴))
36 rnxpid 6125 . . . . 5 ran (𝐴 × 𝐴) = 𝐴
3735, 36sseqtrdi 3994 . . . 4 (𝑅 ⊆ (𝐴 × 𝐴) → ran 𝑅𝐴)
3812, 37unssd 4146 . . 3 (𝑅 ⊆ (𝐴 × 𝐴) → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝐴)
39383ad2ant2 1134 . 2 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝐴)
4034, 39eqssd 3961 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 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  cdif 3907  cun 3908  wss 3910  c0 4282  {csn 4586  cop 4592   class class class wbr 5105   Or wor 5544   × cxp 5631  dom cdm 5633  ran crn 5634  Rel wrel 5638
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-dm 5643  df-rn 5644
This theorem is referenced by: (None)
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