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Theorem suplub 8407
Description: A supremum is the least upper bound. See also supcl 8405 and supub 8406. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 24-Dec-2016.)
Hypotheses
Ref Expression
supmo.1 (𝜑𝑅 Or 𝐴)
supcl.2 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
Assertion
Ref Expression
suplub (𝜑 → ((𝐶𝐴𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧𝐵 𝐶𝑅𝑧))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝑅,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑧,𝐶
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐶(𝑥,𝑦)

Proof of Theorem suplub
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 simpr 476 . . . . . . 7 ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))
2 breq1 4688 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑦𝑅𝑥𝑤𝑅𝑥))
3 breq1 4688 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑦𝑅𝑧𝑤𝑅𝑧))
43rexbidv 3081 . . . . . . . . 9 (𝑦 = 𝑤 → (∃𝑧𝐵 𝑦𝑅𝑧 ↔ ∃𝑧𝐵 𝑤𝑅𝑧))
52, 4imbi12d 333 . . . . . . . 8 (𝑦 = 𝑤 → ((𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)))
65cbvralv 3201 . . . . . . 7 (∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧))
71, 6sylib 208 . . . . . 6 ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧))
87a1i 11 . . . . 5 (𝑥𝐴 → ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)))
98ss2rabi 3717 . . . 4 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} ⊆ {𝑥𝐴 ∣ ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)}
10 supmo.1 . . . . . 6 (𝜑𝑅 Or 𝐴)
1110supval2 8402 . . . . 5 (𝜑 → sup(𝐵, 𝐴, 𝑅) = (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))))
12 supcl.2 . . . . . . 7 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
1310, 12supeu 8401 . . . . . 6 (𝜑 → ∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
14 riotacl2 6664 . . . . . 6 (∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))) ∈ {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))})
1513, 14syl 17 . . . . 5 (𝜑 → (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))) ∈ {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))})
1611, 15eqeltrd 2730 . . . 4 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))})
179, 16sseldi 3634 . . 3 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)})
18 breq2 4689 . . . . . . 7 (𝑥 = sup(𝐵, 𝐴, 𝑅) → (𝑤𝑅𝑥𝑤𝑅sup(𝐵, 𝐴, 𝑅)))
1918imbi1d 330 . . . . . 6 (𝑥 = sup(𝐵, 𝐴, 𝑅) → ((𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧) ↔ (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧)))
2019ralbidv 3015 . . . . 5 (𝑥 = sup(𝐵, 𝐴, 𝑅) → (∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧) ↔ ∀𝑤𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧)))
2120elrab 3396 . . . 4 (sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)} ↔ (sup(𝐵, 𝐴, 𝑅) ∈ 𝐴 ∧ ∀𝑤𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧)))
2221simprbi 479 . . 3 (sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)} → ∀𝑤𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧))
2317, 22syl 17 . 2 (𝜑 → ∀𝑤𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧))
24 breq1 4688 . . . . 5 (𝑤 = 𝐶 → (𝑤𝑅sup(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
25 breq1 4688 . . . . . 6 (𝑤 = 𝐶 → (𝑤𝑅𝑧𝐶𝑅𝑧))
2625rexbidv 3081 . . . . 5 (𝑤 = 𝐶 → (∃𝑧𝐵 𝑤𝑅𝑧 ↔ ∃𝑧𝐵 𝐶𝑅𝑧))
2724, 26imbi12d 333 . . . 4 (𝑤 = 𝐶 → ((𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧) ↔ (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝐶𝑅𝑧)))
2827rspccv 3337 . . 3 (∀𝑤𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧) → (𝐶𝐴 → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝐶𝑅𝑧)))
2928impd 446 . 2 (∀𝑤𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧) → ((𝐶𝐴𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧𝐵 𝐶𝑅𝑧))
3023, 29syl 17 1 (𝜑 → ((𝐶𝐴𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧𝐵 𝐶𝑅𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  ∃!wreu 2943  {crab 2945   class class class wbr 4685   Or wor 5063  crio 6650  supcsup 8387
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
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-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  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-uni 4469  df-br 4686  df-po 5064  df-so 5065  df-iota 5889  df-riota 6651  df-sup 8389
This theorem is referenced by:  suplub2  8408  supnub  8409  supiso  8422  infglb  8437  supxrun  12184  supxrunb1  12187  supxrunb2  12188  esum2d  30283  gtinfOLD  32439
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