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Theorem sofld 5616
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 5160 . . . . . . . . 9 Rel (𝐴 × 𝐴)
2 relss 5240 . . . . . . . . 9 (𝑅 ⊆ (𝐴 × 𝐴) → (Rel (𝐴 × 𝐴) → Rel 𝑅))
31, 2mpi 20 . . . . . . . 8 (𝑅 ⊆ (𝐴 × 𝐴) → Rel 𝑅)
43ad2antlr 763 . . . . . . 7 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → Rel 𝑅)
5 df-br 4686 . . . . . . . . . 10 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
6 ssun1 3809 . . . . . . . . . . . . 13 𝐴 ⊆ (𝐴 ∪ {𝑥})
7 undif1 4076 . . . . . . . . . . . . 13 ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥})
86, 7sseqtr4i 3671 . . . . . . . . . . . 12 𝐴 ⊆ ((𝐴 ∖ {𝑥}) ∪ {𝑥})
9 simpll 805 . . . . . . . . . . . . . 14 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑅 Or 𝐴)
10 dmss 5355 . . . . . . . . . . . . . . . . 17 (𝑅 ⊆ (𝐴 × 𝐴) → dom 𝑅 ⊆ dom (𝐴 × 𝐴))
11 dmxpid 5377 . . . . . . . . . . . . . . . . 17 dom (𝐴 × 𝐴) = 𝐴
1210, 11syl6sseq 3684 . . . . . . . . . . . . . . . 16 (𝑅 ⊆ (𝐴 × 𝐴) → dom 𝑅𝐴)
1312ad2antlr 763 . . . . . . . . . . . . . . 15 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → dom 𝑅𝐴)
143ad2antlr 763 . . . . . . . . . . . . . . . 16 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → Rel 𝑅)
15 releldm 5390 . . . . . . . . . . . . . . . 16 ((Rel 𝑅𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
1614, 15sylancom 702 . . . . . . . . . . . . . . 15 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
1713, 16sseldd 3637 . . . . . . . . . . . . . 14 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥𝐴)
18 sossfld 5615 . . . . . . . . . . . . . 14 ((𝑅 Or 𝐴𝑥𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
199, 17, 18syl2anc 694 . . . . . . . . . . . . 13 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
20 ssun1 3809 . . . . . . . . . . . . . . 15 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2120, 16sseldi 3634 . . . . . . . . . . . . . 14 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅))
2221snssd 4372 . . . . . . . . . . . . 13 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → {𝑥} ⊆ (dom 𝑅 ∪ ran 𝑅))
2319, 22unssd 3822 . . . . . . . . . . . 12 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
248, 23syl5ss 3647 . . . . . . . . . . 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 233 . . . . . . . . 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 118 . . . . . . 7 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ ∅))
294, 28relssdv 5246 . . . . . 6 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → 𝑅 ⊆ ∅)
30 ss0 4007 . . . . . 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 2840 . . 3 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) → (𝑅 ≠ ∅ → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
34333impia 1280 . 2 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅))
35 rnss 5386 . . . . 5 (𝑅 ⊆ (𝐴 × 𝐴) → ran 𝑅 ⊆ ran (𝐴 × 𝐴))
36 rnxpid 5602 . . . . 5 ran (𝐴 × 𝐴) = 𝐴
3735, 36syl6sseq 3684 . . . 4 (𝑅 ⊆ (𝐴 × 𝐴) → ran 𝑅𝐴)
3812, 37unssd 3822 . . 3 (𝑅 ⊆ (𝐴 × 𝐴) → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝐴)
39383ad2ant2 1103 . 2 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝐴)
4034, 39eqssd 3653 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 1054   = wceq 1523  wcel 2030  wne 2823  cdif 3604  cun 3605  wss 3607  c0 3948  {csn 4210  cop 4216   class class class wbr 4685   Or wor 5063   × cxp 5141  dom cdm 5143  ran crn 5144  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153  df-rn 5154
This theorem is referenced by: (None)
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