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Theorem suplub 9454
Description: A supremum is the least upper bound. See also supcl 9452 and supub 9453. (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 484 . . . . . . 7 ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))
2 breq1 5144 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑦𝑅𝑥𝑤𝑅𝑥))
3 breq1 5144 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑦𝑅𝑧𝑤𝑅𝑧))
43rexbidv 3172 . . . . . . . . 9 (𝑦 = 𝑤 → (∃𝑧𝐵 𝑦𝑅𝑧 ↔ ∃𝑧𝐵 𝑤𝑅𝑧))
52, 4imbi12d 344 . . . . . . . 8 (𝑦 = 𝑤 → ((𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)))
65cbvralvw 3228 . . . . . . 7 (∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧))
71, 6sylib 217 . . . . . 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 4069 . . . 4 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} ⊆ {𝑥𝐴 ∣ ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)}
10 supmo.1 . . . . . 6 (𝜑𝑅 Or 𝐴)
1110supval2 9449 . . . . 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 9448 . . . . . 6 (𝜑 → ∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
14 riotacl2 7377 . . . . . 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 2827 . . . 4 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))})
179, 16sselid 3975 . . 3 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)})
18 breq2 5145 . . . . . . 7 (𝑥 = sup(𝐵, 𝐴, 𝑅) → (𝑤𝑅𝑥𝑤𝑅sup(𝐵, 𝐴, 𝑅)))
1918imbi1d 341 . . . . . 6 (𝑥 = sup(𝐵, 𝐴, 𝑅) → ((𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧) ↔ (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧)))
2019ralbidv 3171 . . . . 5 (𝑥 = sup(𝐵, 𝐴, 𝑅) → (∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧) ↔ ∀𝑤𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧)))
2120elrab 3678 . . . 4 (sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)} ↔ (sup(𝐵, 𝐴, 𝑅) ∈ 𝐴 ∧ ∀𝑤𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧)))
2221simprbi 496 . . 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 5144 . . . . 5 (𝑤 = 𝐶 → (𝑤𝑅sup(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
25 breq1 5144 . . . . . 6 (𝑤 = 𝐶 → (𝑤𝑅𝑧𝐶𝑅𝑧))
2625rexbidv 3172 . . . . 5 (𝑤 = 𝐶 → (∃𝑧𝐵 𝑤𝑅𝑧 ↔ ∃𝑧𝐵 𝐶𝑅𝑧))
2724, 26imbi12d 344 . . . 4 (𝑤 = 𝐶 → ((𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧) ↔ (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝐶𝑅𝑧)))
2827rspccv 3603 . . 3 (∀𝑤𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧) → (𝐶𝐴 → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝐶𝑅𝑧)))
2928impd 410 . 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 395   = wceq 1533  wcel 2098  wral 3055  wrex 3064  ∃!wreu 3368  {crab 3426   class class class wbr 5141   Or wor 5580  crio 7359  supcsup 9434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-po 5581  df-so 5582  df-iota 6488  df-riota 7360  df-sup 9436
This theorem is referenced by:  suplub2  9455  supnub  9456  supiso  9469  infglb  9484  supxrun  13298  supxrunb1  13301  supxrunb2  13302  esum2d  33621
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