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Theorem sofld 5481
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 5134 . . . . . . . . 9 Rel (𝐴 × 𝐴)
2 relss 5114 . . . . . . . . 9 (𝑅 ⊆ (𝐴 × 𝐴) → (Rel (𝐴 × 𝐴) → Rel 𝑅))
31, 2mpi 20 . . . . . . . 8 (𝑅 ⊆ (𝐴 × 𝐴) → Rel 𝑅)
43ad2antlr 758 . . . . . . 7 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → Rel 𝑅)
5 df-br 4573 . . . . . . . . . 10 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
6 ssun1 3732 . . . . . . . . . . . . 13 𝐴 ⊆ (𝐴 ∪ {𝑥})
7 undif1 3989 . . . . . . . . . . . . 13 ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥})
86, 7sseqtr4i 3595 . . . . . . . . . . . 12 𝐴 ⊆ ((𝐴 ∖ {𝑥}) ∪ {𝑥})
9 simpll 785 . . . . . . . . . . . . . 14 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑅 Or 𝐴)
10 dmss 5227 . . . . . . . . . . . . . . . . 17 (𝑅 ⊆ (𝐴 × 𝐴) → dom 𝑅 ⊆ dom (𝐴 × 𝐴))
11 dmxpid 5248 . . . . . . . . . . . . . . . . 17 dom (𝐴 × 𝐴) = 𝐴
1210, 11syl6sseq 3608 . . . . . . . . . . . . . . . 16 (𝑅 ⊆ (𝐴 × 𝐴) → dom 𝑅𝐴)
1312ad2antlr 758 . . . . . . . . . . . . . . 15 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → dom 𝑅𝐴)
143ad2antlr 758 . . . . . . . . . . . . . . . 16 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → Rel 𝑅)
15 releldm 5261 . . . . . . . . . . . . . . . 16 ((Rel 𝑅𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
1614, 15sylancom 697 . . . . . . . . . . . . . . 15 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ dom 𝑅)
1713, 16sseldd 3563 . . . . . . . . . . . . . 14 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥𝐴)
18 sossfld 5480 . . . . . . . . . . . . . 14 ((𝑅 Or 𝐴𝑥𝐴) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
199, 17, 18syl2anc 690 . . . . . . . . . . . . 13 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → (𝐴 ∖ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
20 ssun1 3732 . . . . . . . . . . . . . . 15 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2120, 16sseldi 3560 . . . . . . . . . . . . . 14 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝑥 ∈ (dom 𝑅 ∪ ran 𝑅))
2221snssd 4275 . . . . . . . . . . . . 13 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → {𝑥} ⊆ (dom 𝑅 ∪ ran 𝑅))
2319, 22unssd 3745 . . . . . . . . . . . 12 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) ⊆ (dom 𝑅 ∪ ran 𝑅))
248, 23syl5ss 3573 . . . . . . . . . . 11 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ 𝑥𝑅𝑦) → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅))
2524ex 448 . . . . . . . . . 10 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) → (𝑥𝑅𝑦𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
265, 25syl5bir 231 . . . . . . . . 9 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) → (⟨𝑥, 𝑦⟩ ∈ 𝑅𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
2726con3dimp 455 . . . . . . . 8 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → ¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
2827pm2.21d 116 . . . . . . 7 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ ∅))
294, 28relssdv 5119 . . . . . 6 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → 𝑅 ⊆ ∅)
30 ss0 3920 . . . . . 6 (𝑅 ⊆ ∅ → 𝑅 = ∅)
3129, 30syl 17 . . . . 5 (((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) ∧ ¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)) → 𝑅 = ∅)
3231ex 448 . . . 4 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) → (¬ 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅) → 𝑅 = ∅))
3332necon1ad 2793 . . 3 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴)) → (𝑅 ≠ ∅ → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅)))
34333impia 1252 . 2 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → 𝐴 ⊆ (dom 𝑅 ∪ ran 𝑅))
35 rnss 5257 . . . . 5 (𝑅 ⊆ (𝐴 × 𝐴) → ran 𝑅 ⊆ ran (𝐴 × 𝐴))
36 rnxpid 5467 . . . . 5 ran (𝐴 × 𝐴) = 𝐴
3735, 36syl6sseq 3608 . . . 4 (𝑅 ⊆ (𝐴 × 𝐴) → ran 𝑅𝐴)
3812, 37unssd 3745 . . 3 (𝑅 ⊆ (𝐴 × 𝐴) → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝐴)
39383ad2ant2 1075 . 2 ((𝑅 Or 𝐴𝑅 ⊆ (𝐴 × 𝐴) ∧ 𝑅 ≠ ∅) → (dom 𝑅 ∪ ran 𝑅) ⊆ 𝐴)
4034, 39eqssd 3579 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 382  w3a 1030   = wceq 1474  wcel 1975  wne 2774  cdif 3531  cun 3532  wss 3534  c0 3868  {csn 4119  cop 4125   class class class wbr 4572   Or wor 4943   × cxp 5021  dom cdm 5023  ran crn 5024  Rel wrel 5028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pr 4823
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-op 4126  df-br 4573  df-opab 4633  df-po 4944  df-so 4945  df-xp 5029  df-rel 5030  df-cnv 5031  df-dm 5033  df-rn 5034
This theorem is referenced by: (None)
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