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Theorem suplub 8310
Description: A supremum is the least upper bound. See also supcl 8308 and supub 8309. (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 477 . . . . . . 7 ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))
2 breq1 4616 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑦𝑅𝑥𝑤𝑅𝑥))
3 breq1 4616 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑦𝑅𝑧𝑤𝑅𝑧))
43rexbidv 3045 . . . . . . . . 9 (𝑦 = 𝑤 → (∃𝑧𝐵 𝑦𝑅𝑧 ↔ ∃𝑧𝐵 𝑤𝑅𝑧))
52, 4imbi12d 334 . . . . . . . 8 (𝑦 = 𝑤 → ((𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)))
65cbvralv 3159 . . . . . . 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 3663 . . . 4 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} ⊆ {𝑥𝐴 ∣ ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)}
10 supmo.1 . . . . . 6 (𝜑𝑅 Or 𝐴)
1110supval2 8305 . . . . 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 8304 . . . . . 6 (𝜑 → ∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
14 riotacl2 6578 . . . . . 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 2698 . . . 4 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))})
179, 16sseldi 3581 . . 3 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)})
18 breq2 4617 . . . . . . 7 (𝑥 = sup(𝐵, 𝐴, 𝑅) → (𝑤𝑅𝑥𝑤𝑅sup(𝐵, 𝐴, 𝑅)))
1918imbi1d 331 . . . . . 6 (𝑥 = sup(𝐵, 𝐴, 𝑅) → ((𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧) ↔ (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧)))
2019ralbidv 2980 . . . . 5 (𝑥 = sup(𝐵, 𝐴, 𝑅) → (∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧) ↔ ∀𝑤𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧)))
2120elrab 3346 . . . 4 (sup(𝐵, 𝐴, 𝑅) ∈ {𝑥𝐴 ∣ ∀𝑤𝐴 (𝑤𝑅𝑥 → ∃𝑧𝐵 𝑤𝑅𝑧)} ↔ (sup(𝐵, 𝐴, 𝑅) ∈ 𝐴 ∧ ∀𝑤𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧)))
2221simprbi 480 . . 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 4616 . . . . 5 (𝑤 = 𝐶 → (𝑤𝑅sup(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
25 breq1 4616 . . . . . 6 (𝑤 = 𝐶 → (𝑤𝑅𝑧𝐶𝑅𝑧))
2625rexbidv 3045 . . . . 5 (𝑤 = 𝐶 → (∃𝑧𝐵 𝑤𝑅𝑧 ↔ ∃𝑧𝐵 𝐶𝑅𝑧))
2724, 26imbi12d 334 . . . 4 (𝑤 = 𝐶 → ((𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧) ↔ (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝐶𝑅𝑧)))
2827rspccv 3292 . . 3 (∀𝑤𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝑤𝑅𝑧) → (𝐶𝐴 → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧𝐵 𝐶𝑅𝑧)))
2928impd 447 . 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 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  ∃!wreu 2909  {crab 2911   class class class wbr 4613   Or wor 4994  crio 6564  supcsup 8290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-po 4995  df-so 4996  df-iota 5810  df-riota 6565  df-sup 8292
This theorem is referenced by:  suplub2  8311  supnub  8312  supiso  8325  infglb  8340  supxrun  12089  supxrunb1  12092  supxrunb2  12093  esum2d  29936  gtinfOLD  31956
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