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Theorem dfsup2 8625
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 8623 . 2 sup(𝐵, 𝐴, 𝑅) = {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))}
2 dfrab3 4133 . . . 4 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = (𝐴 ∩ {𝑥 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))})
3 abeq1 2938 . . . . . . 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 3417 . . . . . . . . 9 𝑥 ∈ V
5 eldif 3808 . . . . . . . . 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 5716 . . . . . . . . . . . 12 (𝑥 ∈ (𝑅𝐵) ↔ ∃𝑦𝐵 𝑦𝑅𝑥)
8 dfrex2 3204 . . . . . . . . . . . 12 (∃𝑦𝐵 𝑦𝑅𝑥 ↔ ¬ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
97, 8bitri 267 . . . . . . . . . . 11 (𝑥 ∈ (𝑅𝐵) ↔ ¬ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
104elima 5716 . . . . . . . . . . . 12 (𝑥 ∈ (𝑅 “ (𝐴 ∖ (𝑅𝐵))) ↔ ∃𝑦 ∈ (𝐴 ∖ (𝑅𝐵))𝑦𝑅𝑥)
11 dfrex2 3204 . . . . . . . . . . . 12 (∃𝑦 ∈ (𝐴 ∖ (𝑅𝐵))𝑦𝑅𝑥 ↔ ¬ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥)
1210, 11bitri 267 . . . . . . . . . . 11 (𝑥 ∈ (𝑅 “ (𝐴 ∖ (𝑅𝐵))) ↔ ¬ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥)
139, 12orbi12i 943 . . . . . . . . . 10 ((𝑥 ∈ (𝑅𝐵) ∨ 𝑥 ∈ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))) ↔ (¬ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∨ ¬ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥))
14 elun 3982 . . . . . . . . . 10 (𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))) ↔ (𝑥 ∈ (𝑅𝐵) ∨ 𝑥 ∈ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
15 ianor 1009 . . . . . . . . . 10 (¬ (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥) ↔ (¬ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∨ ¬ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥))
1613, 14, 153bitr4i 295 . . . . . . . . 9 (𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))) ↔ ¬ (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥))
1716con2bii 349 . . . . . . . 8 ((∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥) ↔ ¬ 𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
18 vex 3417 . . . . . . . . . . . 12 𝑦 ∈ V
1918, 4brcnv 5541 . . . . . . . . . . 11 (𝑦𝑅𝑥𝑥𝑅𝑦)
2019notbii 312 . . . . . . . . . 10 𝑦𝑅𝑥 ↔ ¬ 𝑥𝑅𝑦)
2120ralbii 3189 . . . . . . . . 9 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦)
22 impexp 443 . . . . . . . . . . 11 (((𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝑅𝐵)) → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝐴 → (¬ 𝑦 ∈ (𝑅𝐵) → ¬ 𝑦𝑅𝑥)))
23 eldif 3808 . . . . . . . . . . . 12 (𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ↔ (𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝑅𝐵)))
2423imbi1i 341 . . . . . . . . . . 11 ((𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) → ¬ 𝑦𝑅𝑥) ↔ ((𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝑅𝐵)) → ¬ 𝑦𝑅𝑥))
2518elima 5716 . . . . . . . . . . . . . . 15 (𝑦 ∈ (𝑅𝐵) ↔ ∃𝑧𝐵 𝑧𝑅𝑦)
26 vex 3417 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
2726, 18brcnv 5541 . . . . . . . . . . . . . . . 16 (𝑧𝑅𝑦𝑦𝑅𝑧)
2827rexbii 3251 . . . . . . . . . . . . . . 15 (∃𝑧𝐵 𝑧𝑅𝑦 ↔ ∃𝑧𝐵 𝑦𝑅𝑧)
2925, 28bitri 267 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝑅𝐵) ↔ ∃𝑧𝐵 𝑦𝑅𝑧)
3029imbi2i 328 . . . . . . . . . . . . 13 ((𝑦𝑅𝑥𝑦 ∈ (𝑅𝐵)) ↔ (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))
31 con34b 308 . . . . . . . . . . . . 13 ((𝑦𝑅𝑥𝑦 ∈ (𝑅𝐵)) ↔ (¬ 𝑦 ∈ (𝑅𝐵) → ¬ 𝑦𝑅𝑥))
3230, 31bitr3i 269 . . . . . . . . . . . 12 ((𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ (¬ 𝑦 ∈ (𝑅𝐵) → ¬ 𝑦𝑅𝑥))
3332imbi2i 328 . . . . . . . . . . 11 ((𝑦𝐴 → (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ↔ (𝑦𝐴 → (¬ 𝑦 ∈ (𝑅𝐵) → ¬ 𝑦𝑅𝑥)))
3422, 24, 333bitr4i 295 . . . . . . . . . 10 ((𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝐴 → (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
3534ralbii2 3187 . . . . . . . . 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 292 . . . . . . 7 ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ↔ 𝑥 ∈ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))))
383, 37mpgbir 1898 . . . . . 6 {𝑥 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
3938ineq2i 4040 . . . . 5 (𝐴 ∩ {𝑥 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))}) = (𝐴 ∩ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))))
40 invdif 4100 . . . . 5 (𝐴 ∩ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))) = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
4139, 40eqtri 2849 . . . 4 (𝐴 ∩ {𝑥 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))}) = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
422, 41eqtri 2849 . . 3 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
4342unieqi 4669 . 2 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
441, 43eqtri 2849 1 sup(𝐵, 𝐴, 𝑅) = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wo 878   = wceq 1656  wcel 2164  {cab 2811  wral 3117  wrex 3118  {crab 3121  Vcvv 3414  cdif 3795  cun 3796  cin 3797   cuni 4660   class class class wbr 4875  ccnv 5345  cima 5349  supcsup 8621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-xp 5352  df-cnv 5354  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-sup 8623
This theorem is referenced by:  nfsup  8632
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