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Theorem dfsup2 8896
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 8894 . 2 sup(𝐵, 𝐴, 𝑅) = {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))}
2 dfrab3 4275 . . . 4 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = (𝐴 ∩ {𝑥 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))})
3 abeq1 2943 . . . . . . 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 3495 . . . . . . . . 9 𝑥 ∈ V
5 eldif 3943 . . . . . . . . 9 (𝑥 ∈ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))))
64, 5mpbiran 705 . . . . . . . 8 (𝑥 ∈ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))) ↔ ¬ 𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
74elima 5927 . . . . . . . . . . . 12 (𝑥 ∈ (𝑅𝐵) ↔ ∃𝑦𝐵 𝑦𝑅𝑥)
8 dfrex2 3236 . . . . . . . . . . . 12 (∃𝑦𝐵 𝑦𝑅𝑥 ↔ ¬ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
97, 8bitri 276 . . . . . . . . . . 11 (𝑥 ∈ (𝑅𝐵) ↔ ¬ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
104elima 5927 . . . . . . . . . . . 12 (𝑥 ∈ (𝑅 “ (𝐴 ∖ (𝑅𝐵))) ↔ ∃𝑦 ∈ (𝐴 ∖ (𝑅𝐵))𝑦𝑅𝑥)
11 dfrex2 3236 . . . . . . . . . . . 12 (∃𝑦 ∈ (𝐴 ∖ (𝑅𝐵))𝑦𝑅𝑥 ↔ ¬ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥)
1210, 11bitri 276 . . . . . . . . . . 11 (𝑥 ∈ (𝑅 “ (𝐴 ∖ (𝑅𝐵))) ↔ ¬ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥)
139, 12orbi12i 908 . . . . . . . . . 10 ((𝑥 ∈ (𝑅𝐵) ∨ 𝑥 ∈ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))) ↔ (¬ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∨ ¬ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥))
14 elun 4122 . . . . . . . . . 10 (𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))) ↔ (𝑥 ∈ (𝑅𝐵) ∨ 𝑥 ∈ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
15 ianor 975 . . . . . . . . . 10 (¬ (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥) ↔ (¬ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∨ ¬ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥))
1613, 14, 153bitr4i 304 . . . . . . . . 9 (𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))) ↔ ¬ (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥))
1716con2bii 359 . . . . . . . 8 ((∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥) ↔ ¬ 𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
18 vex 3495 . . . . . . . . . . . 12 𝑦 ∈ V
1918, 4brcnv 5746 . . . . . . . . . . 11 (𝑦𝑅𝑥𝑥𝑅𝑦)
2019notbii 321 . . . . . . . . . 10 𝑦𝑅𝑥 ↔ ¬ 𝑥𝑅𝑦)
2120ralbii 3162 . . . . . . . . 9 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦)
22 impexp 451 . . . . . . . . . . 11 (((𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝑅𝐵)) → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝐴 → (¬ 𝑦 ∈ (𝑅𝐵) → ¬ 𝑦𝑅𝑥)))
23 eldif 3943 . . . . . . . . . . . 12 (𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ↔ (𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝑅𝐵)))
2423imbi1i 351 . . . . . . . . . . 11 ((𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) → ¬ 𝑦𝑅𝑥) ↔ ((𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝑅𝐵)) → ¬ 𝑦𝑅𝑥))
2518elima 5927 . . . . . . . . . . . . . . 15 (𝑦 ∈ (𝑅𝐵) ↔ ∃𝑧𝐵 𝑧𝑅𝑦)
26 vex 3495 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
2726, 18brcnv 5746 . . . . . . . . . . . . . . . 16 (𝑧𝑅𝑦𝑦𝑅𝑧)
2827rexbii 3244 . . . . . . . . . . . . . . 15 (∃𝑧𝐵 𝑧𝑅𝑦 ↔ ∃𝑧𝐵 𝑦𝑅𝑧)
2925, 28bitri 276 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝑅𝐵) ↔ ∃𝑧𝐵 𝑦𝑅𝑧)
3029imbi2i 337 . . . . . . . . . . . . 13 ((𝑦𝑅𝑥𝑦 ∈ (𝑅𝐵)) ↔ (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))
31 con34b 317 . . . . . . . . . . . . 13 ((𝑦𝑅𝑥𝑦 ∈ (𝑅𝐵)) ↔ (¬ 𝑦 ∈ (𝑅𝐵) → ¬ 𝑦𝑅𝑥))
3230, 31bitr3i 278 . . . . . . . . . . . 12 ((𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ (¬ 𝑦 ∈ (𝑅𝐵) → ¬ 𝑦𝑅𝑥))
3332imbi2i 337 . . . . . . . . . . 11 ((𝑦𝐴 → (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ↔ (𝑦𝐴 → (¬ 𝑦 ∈ (𝑅𝐵) → ¬ 𝑦𝑅𝑥)))
3422, 24, 333bitr4i 304 . . . . . . . . . 10 ((𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝐴 → (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
3534ralbii2 3160 . . . . . . . . 9 (∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))
3621, 35anbi12i 626 . . . . . . . 8 ((∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥) ↔ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
376, 17, 363bitr2ri 301 . . . . . . 7 ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ↔ 𝑥 ∈ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))))
383, 37mpgbir 1791 . . . . . 6 {𝑥 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
3938ineq2i 4183 . . . . 5 (𝐴 ∩ {𝑥 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))}) = (𝐴 ∩ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))))
40 invdif 4242 . . . . 5 (𝐴 ∩ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))) = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
4139, 40eqtri 2841 . . . 4 (𝐴 ∩ {𝑥 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))}) = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
422, 41eqtri 2841 . . 3 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
4342unieqi 4839 . 2 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
441, 43eqtri 2841 1 sup(𝐵, 𝐴, 𝑅) = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  {cab 2796  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  cdif 3930  cun 3931  cin 3932   cuni 4830   class class class wbr 5057  ccnv 5547  cima 5551  supcsup 8892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-sup 8894
This theorem is referenced by:  nfsup  8903
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