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Theorem suplubti 6887
Description: A supremum is the least upper bound. See also supclti 6885 and supubti 6886. (Contributed by Jim Kingdon, 24-Nov-2021.)
Hypotheses
Ref Expression
supmoti.ti ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
supclti.2 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
Assertion
Ref Expression
suplubti (𝜑 → ((𝐶𝐴𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧𝐵 𝐶𝑅𝑧))
Distinct variable groups:   𝑢,𝐴,𝑣,𝑥   𝑦,𝐴,𝑥,𝑧   𝑥,𝐵,𝑦,𝑧   𝑢,𝑅,𝑣,𝑥   𝑦,𝑅,𝑧   𝜑,𝑢,𝑣,𝑥   𝑧,𝐶
Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐵(𝑣,𝑢)   𝐶(𝑥,𝑦,𝑣,𝑢)

Proof of Theorem suplubti
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 simpr 109 . . . . . 6 ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))
2 breq1 3932 . . . . . . . 8 (𝑦 = 𝑤 → (𝑦𝑅𝑥𝑤𝑅𝑥))
3 breq1 3932 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑦𝑅𝑧𝑤𝑅𝑧))
43rexbidv 2438 . . . . . . . 8 (𝑦 = 𝑤 → (∃𝑧𝐵 𝑦𝑅𝑧 ↔ ∃𝑧𝐵 𝑤𝑅𝑧))
52, 4imbi12d 233 . . . . . . 7 (𝑦 = 𝑤 → ((𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)))
65cbvralv 2654 . . . . . 6 (∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧))
71, 6sylib 121 . . . . 5 ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧))
87a1i 9 . . . 4 (𝑥𝐴 → ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)))
98ss2rabi 3179 . . 3 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} ⊆ {𝑥𝐴 ∣ ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)}
10 supmoti.ti . . . . 5 ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
11 supclti.2 . . . . 5 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
1210, 11supval2ti 6882 . . . 4 (𝜑 → sup(𝐵, 𝐴, 𝑅) = (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))))
1310, 11supeuti 6881 . . . . 5 (𝜑 → ∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
14 riotacl2 5743 . . . . 5 (∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))) ∈ {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))})
1513, 14syl 14 . . . 4 (𝜑 → (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))) ∈ {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))})
1612, 15eqeltrd 2216 . . 3 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))})
179, 16sseldi 3095 . 2 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)})
18 breq2 3933 . . . . . 6 (𝑥 = sup(𝐵, 𝐴, 𝑅) → (𝑤𝑅𝑥𝑤𝑅sup(𝐵, 𝐴, 𝑅)))
1918imbi1d 230 . . . . 5 (𝑥 = sup(𝐵, 𝐴, 𝑅) → ((𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧) ↔ (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧)))
2019ralbidv 2437 . . . 4 (𝑥 = sup(𝐵, 𝐴, 𝑅) → (∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧) ↔ ∀𝑤𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧)))
2120elrab 2840 . . 3 (sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)} ↔ (sup(𝐵, 𝐴, 𝑅) ∈ 𝐴 ∧ ∀𝑤𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧)))
2221simprbi 273 . 2 (sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)} → ∀𝑤𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧))
23 breq1 3932 . . . . 5 (𝑤 = 𝐶 → (𝑤𝑅sup(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
24 breq1 3932 . . . . . 6 (𝑤 = 𝐶 → (𝑤𝑅𝑧𝐶𝑅𝑧))
2524rexbidv 2438 . . . . 5 (𝑤 = 𝐶 → (∃𝑧𝐵 𝑤𝑅𝑧 ↔ ∃𝑧𝐵 𝐶𝑅𝑧))
2623, 25imbi12d 233 . . . 4 (𝑤 = 𝐶 → ((𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧) ↔ (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝐶𝑅𝑧)))
2726rspccv 2786 . . 3 (∀𝑤𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧) → (𝐶𝐴 → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝐶𝑅𝑧)))
2827impd 252 . 2 (∀𝑤𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧) → ((𝐶𝐴𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧𝐵 𝐶𝑅𝑧))
2917, 22, 283syl 17 1 (𝜑 → ((𝐶𝐴𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧𝐵 𝐶𝑅𝑧))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  wral 2416  wrex 2417  ∃!wreu 2418  {crab 2420   class class class wbr 3929  crio 5729  supcsup 6869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-riota 5730  df-sup 6871
This theorem is referenced by:  suplub2ti  6888  supisoti  6897  infglbti  6912  sup3exmid  8722  maxleast  10992
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