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Theorem dfsup2 9481
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 9479 . 2 sup(𝐵, 𝐴, 𝑅) = {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))}
2 dfrab3 4324 . . . 4 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = (𝐴 ∩ {𝑥 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))})
3 eqabcb 2880 . . . . . . 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 3481 . . . . . . . . 9 𝑥 ∈ V
5 eldif 3972 . . . . . . . . 9 (𝑥 ∈ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))))
64, 5mpbiran 709 . . . . . . . 8 (𝑥 ∈ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))) ↔ ¬ 𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
74elima 6084 . . . . . . . . . . . 12 (𝑥 ∈ (𝑅𝐵) ↔ ∃𝑦𝐵 𝑦𝑅𝑥)
8 dfrex2 3070 . . . . . . . . . . . 12 (∃𝑦𝐵 𝑦𝑅𝑥 ↔ ¬ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
97, 8bitri 275 . . . . . . . . . . 11 (𝑥 ∈ (𝑅𝐵) ↔ ¬ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥)
104elima 6084 . . . . . . . . . . . 12 (𝑥 ∈ (𝑅 “ (𝐴 ∖ (𝑅𝐵))) ↔ ∃𝑦 ∈ (𝐴 ∖ (𝑅𝐵))𝑦𝑅𝑥)
11 dfrex2 3070 . . . . . . . . . . . 12 (∃𝑦 ∈ (𝐴 ∖ (𝑅𝐵))𝑦𝑅𝑥 ↔ ¬ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥)
1210, 11bitri 275 . . . . . . . . . . 11 (𝑥 ∈ (𝑅 “ (𝐴 ∖ (𝑅𝐵))) ↔ ¬ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥)
139, 12orbi12i 914 . . . . . . . . . 10 ((𝑥 ∈ (𝑅𝐵) ∨ 𝑥 ∈ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))) ↔ (¬ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∨ ¬ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥))
14 elun 4162 . . . . . . . . . 10 (𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))) ↔ (𝑥 ∈ (𝑅𝐵) ∨ 𝑥 ∈ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
15 ianor 983 . . . . . . . . . 10 (¬ (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥) ↔ (¬ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∨ ¬ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥))
1613, 14, 153bitr4i 303 . . . . . . . . 9 (𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))) ↔ ¬ (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥))
1716con2bii 357 . . . . . . . 8 ((∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ¬ 𝑦𝑅𝑥) ↔ ¬ 𝑥 ∈ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
18 vex 3481 . . . . . . . . . . . 12 𝑦 ∈ V
1918, 4brcnv 5895 . . . . . . . . . . 11 (𝑦𝑅𝑥𝑥𝑅𝑦)
2019notbii 320 . . . . . . . . . 10 𝑦𝑅𝑥 ↔ ¬ 𝑥𝑅𝑦)
2120ralbii 3090 . . . . . . . . 9 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐵 ¬ 𝑥𝑅𝑦)
22 impexp 450 . . . . . . . . . . 11 (((𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝑅𝐵)) → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝐴 → (¬ 𝑦 ∈ (𝑅𝐵) → ¬ 𝑦𝑅𝑥)))
23 eldif 3972 . . . . . . . . . . . 12 (𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) ↔ (𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝑅𝐵)))
2423imbi1i 349 . . . . . . . . . . 11 ((𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) → ¬ 𝑦𝑅𝑥) ↔ ((𝑦𝐴 ∧ ¬ 𝑦 ∈ (𝑅𝐵)) → ¬ 𝑦𝑅𝑥))
2518elima 6084 . . . . . . . . . . . . . . 15 (𝑦 ∈ (𝑅𝐵) ↔ ∃𝑧𝐵 𝑧𝑅𝑦)
26 vex 3481 . . . . . . . . . . . . . . . . 17 𝑧 ∈ V
2726, 18brcnv 5895 . . . . . . . . . . . . . . . 16 (𝑧𝑅𝑦𝑦𝑅𝑧)
2827rexbii 3091 . . . . . . . . . . . . . . 15 (∃𝑧𝐵 𝑧𝑅𝑦 ↔ ∃𝑧𝐵 𝑦𝑅𝑧)
2925, 28bitri 275 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝑅𝐵) ↔ ∃𝑧𝐵 𝑦𝑅𝑧)
3029imbi2i 336 . . . . . . . . . . . . 13 ((𝑦𝑅𝑥𝑦 ∈ (𝑅𝐵)) ↔ (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))
31 con34b 316 . . . . . . . . . . . . 13 ((𝑦𝑅𝑥𝑦 ∈ (𝑅𝐵)) ↔ (¬ 𝑦 ∈ (𝑅𝐵) → ¬ 𝑦𝑅𝑥))
3230, 31bitr3i 277 . . . . . . . . . . . 12 ((𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ (¬ 𝑦 ∈ (𝑅𝐵) → ¬ 𝑦𝑅𝑥))
3332imbi2i 336 . . . . . . . . . . 11 ((𝑦𝐴 → (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ↔ (𝑦𝐴 → (¬ 𝑦 ∈ (𝑅𝐵) → ¬ 𝑦𝑅𝑥)))
3422, 24, 333bitr4i 303 . . . . . . . . . 10 ((𝑦 ∈ (𝐴 ∖ (𝑅𝐵)) → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝐴 → (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
3534ralbii2 3086 . . . . . . . . 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 300 . . . . . . 7 ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ↔ 𝑥 ∈ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))))
383, 37mpgbir 1795 . . . . . 6 {𝑥 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
3938ineq2i 4224 . . . . 5 (𝐴 ∩ {𝑥 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))}) = (𝐴 ∩ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵))))))
40 invdif 4284 . . . . 5 (𝐴 ∩ (V ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))) = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
4139, 40eqtri 2762 . . . 4 (𝐴 ∩ {𝑥 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))}) = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
422, 41eqtri 2762 . . 3 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
4342unieqi 4923 . 2 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
441, 43eqtri 2762 1 sup(𝐵, 𝐴, 𝑅) = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1536  wcel 2105  {cab 2711  wral 3058  wrex 3067  {crab 3432  Vcvv 3477  cdif 3959  cun 3960  cin 3961   cuni 4911   class class class wbr 5147  ccnv 5687  cima 5691  supcsup 9477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-xp 5694  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-sup 9479
This theorem is referenced by:  nfsup  9488
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