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Theorem dfsup2 8557
Description: Quantifier free definition of supremum. (Contributed by Scott Fenton, 19-Feb-2013.)
Assertion
Ref Expression
dfsup2 sup(𝐵, 𝐴, 𝑅) = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))

Proof of Theorem dfsup2
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sup 8555 . 2 sup(𝐵, 𝐴, 𝑅) = {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))}
2 dfrab3 4066 . . . 4 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = (𝐴 ∩ {𝑥 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))})
3 abeq1 2876 . . . . . . 7 ({𝑥 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))) ↔ ∀𝑥((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ↔ 𝑥 ∈ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))))
4 vex 3353 . . . . . . . . 9 𝑥 ∈ V
5 eldif 3742 . . . . . . . . 9 (𝑥 ∈ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))))
64, 5mpbiran 700 . . . . . . . 8 (𝑥 ∈ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))) ↔ ¬ 𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
74elima 5653 . . . . . . . . . . . 12 (𝑥 ∈ (𝑅𝐵) ↔ ∃𝑦𝐵 𝑦𝑅𝑥)
8 dfrex2 3142 . . . . . . . . . . . 12 (∃𝑦𝐵 𝑦𝑅𝑥 ↔ ¬ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
97, 8bitri 266 . . . . . . . . . . 11 (𝑥 ∈ (𝑅𝐵) ↔ ¬ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
104elima 5653 . . . . . . . . . . . 12 (𝑥 ∈ (𝑅 “ (𝐴 ∖ (𝑅𝐵))) ↔ ∃𝑦 ∈ (𝐴 ∖ (𝑅𝐵))𝑦𝑅𝑥)
11 dfrex2 3142 . . . . . . . . . . . 12 (∃𝑦 ∈ (𝐴 ∖ (𝑅𝐵))𝑦𝑅𝑥 ↔ ¬ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥)
1210, 11bitri 266 . . . . . . . . . . 11 (𝑥 ∈ (𝑅 “ (𝐴 ∖ (𝑅𝐵))) ↔ ¬ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥)
139, 12orbi12i 938 . . . . . . . . . 10 ((𝑥 ∈ (𝑅𝐵) ∨ 𝑥 ∈ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))) ↔ (¬ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∨ ¬ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥))
14 elun 3915 . . . . . . . . . 10 (𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))) ↔ (𝑥 ∈ (𝑅𝐵) ∨ 𝑥 ∈ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
15 ianor 1004 . . . . . . . . . 10 (¬ (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥) ↔ (¬ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∨ ¬ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥))
1613, 14, 153bitr4i 294 . . . . . . . . 9 (𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))) ↔ ¬ (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥))
1716con2bii 348 . . . . . . . 8 ((∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥) ↔ ¬ 𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
18 vex 3353 . . . . . . . . . . . 12 𝑦 ∈ V
1918, 4brcnv 5473 . . . . . . . . . . 11 (𝑦𝑅𝑥𝑥𝑅𝑦)
2019notbii 311 . . . . . . . . . 10 𝑦𝑅𝑥 ↔ ¬ 𝑥𝑅𝑦)
2120ralbii 3127 . . . . . . . . 9 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦)
22 impexp 441 . . . . . . . . . . 11 (((𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝑅𝐵)) → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝐴 → (¬ 𝑦 ∈ (𝑅𝐵) → ¬ 𝑦𝑅𝑥)))
23 eldif 3742 . . . . . . . . . . . 12 (𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ↔ (𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝑅𝐵)))
2423imbi1i 340 . . . . . . . . . . 11 ((𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) → ¬ 𝑦𝑅𝑥) ↔ ((𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝑅𝐵)) → ¬ 𝑦𝑅𝑥))
2518elima 5653 . . . . . . . . . . . . . . 15 (𝑦 ∈ (𝑅𝐵) ↔ ∃𝑧𝐵 𝑧𝑅𝑦)
26 vex 3353 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
2726, 18brcnv 5473 . . . . . . . . . . . . . . . 16 (𝑧𝑅𝑦𝑦𝑅𝑧)
2827rexbii 3188 . . . . . . . . . . . . . . 15 (∃𝑧𝐵 𝑧𝑅𝑦 ↔ ∃𝑧𝐵 𝑦𝑅𝑧)
2925, 28bitri 266 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝑅𝐵) ↔ ∃𝑧𝐵 𝑦𝑅𝑧)
3029imbi2i 327 . . . . . . . . . . . . 13 ((𝑦𝑅𝑥𝑦 ∈ (𝑅𝐵)) ↔ (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))
31 con34b 307 . . . . . . . . . . . . 13 ((𝑦𝑅𝑥𝑦 ∈ (𝑅𝐵)) ↔ (¬ 𝑦 ∈ (𝑅𝐵) → ¬ 𝑦𝑅𝑥))
3230, 31bitr3i 268 . . . . . . . . . . . 12 ((𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ (¬ 𝑦 ∈ (𝑅𝐵) → ¬ 𝑦𝑅𝑥))
3332imbi2i 327 . . . . . . . . . . 11 ((𝑦𝐴 → (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ↔ (𝑦𝐴 → (¬ 𝑦 ∈ (𝑅𝐵) → ¬ 𝑦𝑅𝑥)))
3422, 24, 333bitr4i 294 . . . . . . . . . 10 ((𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝐴 → (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
3534ralbii2 3125 . . . . . . . . 9 (∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))
3621, 35anbi12i 620 . . . . . . . 8 ((∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥) ↔ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
376, 17, 363bitr2ri 291 . . . . . . 7 ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ↔ 𝑥 ∈ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))))
383, 37mpgbir 1894 . . . . . 6 {𝑥 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
3938ineq2i 3973 . . . . 5 (𝐴 ∩ {𝑥 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))}) = (𝐴 ∩ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))))
40 invdif 4033 . . . . 5 (𝐴 ∩ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))) = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
4139, 40eqtri 2787 . . . 4 (𝐴 ∩ {𝑥 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))}) = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
422, 41eqtri 2787 . . 3 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
4342unieqi 4603 . 2 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
441, 43eqtri 2787 1 sup(𝐵, 𝐴, 𝑅) = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873   = wceq 1652  wcel 2155  {cab 2751  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  cdif 3729  cun 3730  cin 3731   cuni 4594   class class class wbr 4809  ccnv 5276  cima 5280  supcsup 8553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-xp 5283  df-cnv 5285  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-sup 8555
This theorem is referenced by:  nfsup  8564
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