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Theorem suplub 8910
Description: A supremum is the least upper bound. See also supcl 8908 and supub 8909. (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 488 . . . . . . 7 ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))
2 breq1 5052 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑦𝑅𝑥𝑤𝑅𝑥))
3 breq1 5052 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑦𝑅𝑧𝑤𝑅𝑧))
43rexbidv 3289 . . . . . . . . 9 (𝑦 = 𝑤 → (∃𝑧𝐵 𝑦𝑅𝑧 ↔ ∃𝑧𝐵 𝑤𝑅𝑧))
52, 4imbi12d 348 . . . . . . . 8 (𝑦 = 𝑤 → ((𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)))
65cbvralvw 3434 . . . . . . 7 (∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧))
71, 6sylib 221 . . . . . 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 4037 . . . 4 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} ⊆ {𝑥𝐴 ∣ ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)}
10 supmo.1 . . . . . 6 (𝜑𝑅 Or 𝐴)
1110supval2 8905 . . . . 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 8904 . . . . . 6 (𝜑 → ∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
14 riotacl2 7114 . . . . . 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 2916 . . . 4 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))})
179, 16sseldi 3949 . . 3 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)})
18 breq2 5053 . . . . . . 7 (𝑥 = sup(𝐵, 𝐴, 𝑅) → (𝑤𝑅𝑥𝑤𝑅sup(𝐵, 𝐴, 𝑅)))
1918imbi1d 345 . . . . . 6 (𝑥 = sup(𝐵, 𝐴, 𝑅) → ((𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧) ↔ (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧)))
2019ralbidv 3191 . . . . 5 (𝑥 = sup(𝐵, 𝐴, 𝑅) → (∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧) ↔ ∀𝑤𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧)))
2120elrab 3665 . . . 4 (sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)} ↔ (sup(𝐵, 𝐴, 𝑅) ∈ 𝐴 ∧ ∀𝑤𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧)))
2221simprbi 500 . . 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 5052 . . . . 5 (𝑤 = 𝐶 → (𝑤𝑅sup(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
25 breq1 5052 . . . . . 6 (𝑤 = 𝐶 → (𝑤𝑅𝑧𝐶𝑅𝑧))
2625rexbidv 3289 . . . . 5 (𝑤 = 𝐶 → (∃𝑧𝐵 𝑤𝑅𝑧 ↔ ∃𝑧𝐵 𝐶𝑅𝑧))
2724, 26imbi12d 348 . . . 4 (𝑤 = 𝐶 → ((𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧) ↔ (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝐶𝑅𝑧)))
2827rspccv 3605 . . 3 (∀𝑤𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧) → (𝐶𝐴 → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝐶𝑅𝑧)))
2928impd 414 . 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 399   = wceq 1538  wcel 2115  wral 3132  wrex 3133  ∃!wreu 3134  {crab 3136   class class class wbr 5049   Or wor 5456  crio 7097  supcsup 8890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-br 5050  df-po 5457  df-so 5458  df-iota 6297  df-riota 7098  df-sup 8892
This theorem is referenced by:  suplub2  8911  supnub  8912  supiso  8925  infglb  8940  supxrun  12697  supxrunb1  12700  supxrunb2  12701  esum2d  31372
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