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Theorem dfsup2 8882
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 8880 . 2 sup(𝐵, 𝐴, 𝑅) = {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))}
2 dfrab3 4252 . . . 4 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = (𝐴 ∩ {𝑥 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))})
3 abeq1 2944 . . . . . . 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 3473 . . . . . . . . 9 𝑥 ∈ V
5 eldif 3919 . . . . . . . . 9 (𝑥 ∈ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))))
64, 5mpbiran 707 . . . . . . . 8 (𝑥 ∈ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))) ↔ ¬ 𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
74elima 5906 . . . . . . . . . . . 12 (𝑥 ∈ (𝑅𝐵) ↔ ∃𝑦𝐵 𝑦𝑅𝑥)
8 dfrex2 3226 . . . . . . . . . . . 12 (∃𝑦𝐵 𝑦𝑅𝑥 ↔ ¬ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
97, 8bitri 277 . . . . . . . . . . 11 (𝑥 ∈ (𝑅𝐵) ↔ ¬ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
104elima 5906 . . . . . . . . . . . 12 (𝑥 ∈ (𝑅 “ (𝐴 ∖ (𝑅𝐵))) ↔ ∃𝑦 ∈ (𝐴 ∖ (𝑅𝐵))𝑦𝑅𝑥)
11 dfrex2 3226 . . . . . . . . . . . 12 (∃𝑦 ∈ (𝐴 ∖ (𝑅𝐵))𝑦𝑅𝑥 ↔ ¬ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥)
1210, 11bitri 277 . . . . . . . . . . 11 (𝑥 ∈ (𝑅 “ (𝐴 ∖ (𝑅𝐵))) ↔ ¬ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥)
139, 12orbi12i 911 . . . . . . . . . 10 ((𝑥 ∈ (𝑅𝐵) ∨ 𝑥 ∈ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))) ↔ (¬ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∨ ¬ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥))
14 elun 4100 . . . . . . . . . 10 (𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))) ↔ (𝑥 ∈ (𝑅𝐵) ∨ 𝑥 ∈ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
15 ianor 978 . . . . . . . . . 10 (¬ (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥) ↔ (¬ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∨ ¬ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥))
1613, 14, 153bitr4i 305 . . . . . . . . 9 (𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))) ↔ ¬ (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥))
1716con2bii 360 . . . . . . . 8 ((∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥) ↔ ¬ 𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
18 vex 3473 . . . . . . . . . . . 12 𝑦 ∈ V
1918, 4brcnv 5725 . . . . . . . . . . 11 (𝑦𝑅𝑥𝑥𝑅𝑦)
2019notbii 322 . . . . . . . . . 10 𝑦𝑅𝑥 ↔ ¬ 𝑥𝑅𝑦)
2120ralbii 3152 . . . . . . . . 9 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦)
22 impexp 453 . . . . . . . . . . 11 (((𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝑅𝐵)) → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝐴 → (¬ 𝑦 ∈ (𝑅𝐵) → ¬ 𝑦𝑅𝑥)))
23 eldif 3919 . . . . . . . . . . . 12 (𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ↔ (𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝑅𝐵)))
2423imbi1i 352 . . . . . . . . . . 11 ((𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) → ¬ 𝑦𝑅𝑥) ↔ ((𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝑅𝐵)) → ¬ 𝑦𝑅𝑥))
2518elima 5906 . . . . . . . . . . . . . . 15 (𝑦 ∈ (𝑅𝐵) ↔ ∃𝑧𝐵 𝑧𝑅𝑦)
26 vex 3473 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
2726, 18brcnv 5725 . . . . . . . . . . . . . . . 16 (𝑧𝑅𝑦𝑦𝑅𝑧)
2827rexbii 3234 . . . . . . . . . . . . . . 15 (∃𝑧𝐵 𝑧𝑅𝑦 ↔ ∃𝑧𝐵 𝑦𝑅𝑧)
2925, 28bitri 277 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝑅𝐵) ↔ ∃𝑧𝐵 𝑦𝑅𝑧)
3029imbi2i 338 . . . . . . . . . . . . 13 ((𝑦𝑅𝑥𝑦 ∈ (𝑅𝐵)) ↔ (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))
31 con34b 318 . . . . . . . . . . . . 13 ((𝑦𝑅𝑥𝑦 ∈ (𝑅𝐵)) ↔ (¬ 𝑦 ∈ (𝑅𝐵) → ¬ 𝑦𝑅𝑥))
3230, 31bitr3i 279 . . . . . . . . . . . 12 ((𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ (¬ 𝑦 ∈ (𝑅𝐵) → ¬ 𝑦𝑅𝑥))
3332imbi2i 338 . . . . . . . . . . 11 ((𝑦𝐴 → (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ↔ (𝑦𝐴 → (¬ 𝑦 ∈ (𝑅𝐵) → ¬ 𝑦𝑅𝑥)))
3422, 24, 333bitr4i 305 . . . . . . . . . 10 ((𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝐴 → (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
3534ralbii2 3150 . . . . . . . . 9 (∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))
3621, 35anbi12i 628 . . . . . . . 8 ((∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥) ↔ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
376, 17, 363bitr2ri 302 . . . . . . 7 ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ↔ 𝑥 ∈ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))))
383, 37mpgbir 1800 . . . . . 6 {𝑥 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
3938ineq2i 4160 . . . . 5 (𝐴 ∩ {𝑥 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))}) = (𝐴 ∩ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))))
40 invdif 4219 . . . . 5 (𝐴 ∩ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))) = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
4139, 40eqtri 2843 . . . 4 (𝐴 ∩ {𝑥 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))}) = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
422, 41eqtri 2843 . . 3 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
4342unieqi 4823 . 2 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
441, 43eqtri 2843 1 sup(𝐵, 𝐴, 𝑅) = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843   = wceq 1537  wcel 2114  {cab 2798  wral 3125  wrex 3126  {crab 3129  Vcvv 3470  cdif 3906  cun 3907  cin 3908   cuni 4810   class class class wbr 5038  ccnv 5526  cima 5530  supcsup 8878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pr 5302
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-xp 5533  df-cnv 5535  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-sup 8880
This theorem is referenced by:  nfsup  8889
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