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Theorem infvalti 7052
Description: Alternate expression for the infimum. (Contributed by Jim Kingdon, 17-Dec-2021.)
Hypotheses
Ref Expression
eqinfti.ti ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
infvalti.ex (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
Assertion
Ref Expression
infvalti (𝜑 → inf(𝐵, 𝐴, 𝑅) = (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦))))
Distinct variable groups:   𝑢,𝐴,𝑣,𝑦,𝑧   𝜑,𝑢,𝑣   𝑢,𝑅,𝑣,𝑦,𝑧   𝑢,𝐵,𝑣,𝑦,𝑧   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝜑,𝑥,𝑦,𝑧,𝑢,𝑣

Proof of Theorem infvalti
StepHypRef Expression
1 df-inf 7015 . 2 inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, 𝑅)
2 eqinfti.ti . . . . 5 ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
32cnvti 7049 . . . 4 ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
4 infvalti.ex . . . . 5 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
54cnvinfex 7048 . . . 4 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
63, 5supval2ti 7025 . . 3 (𝜑 → sup(𝐵, 𝐴, 𝑅) = (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))))
7 vex 2755 . . . . . . . . 9 𝑥 ∈ V
8 vex 2755 . . . . . . . . 9 𝑦 ∈ V
97, 8brcnv 4828 . . . . . . . 8 (𝑥𝑅𝑦𝑦𝑅𝑥)
109a1i 9 . . . . . . 7 (𝜑 → (𝑥𝑅𝑦𝑦𝑅𝑥))
1110notbid 668 . . . . . 6 (𝜑 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑦𝑅𝑥))
1211ralbidv 2490 . . . . 5 (𝜑 → (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥))
138, 7brcnv 4828 . . . . . . . 8 (𝑦𝑅𝑥𝑥𝑅𝑦)
1413a1i 9 . . . . . . 7 (𝜑 → (𝑦𝑅𝑥𝑥𝑅𝑦))
15 vex 2755 . . . . . . . . . 10 𝑧 ∈ V
168, 15brcnv 4828 . . . . . . . . 9 (𝑦𝑅𝑧𝑧𝑅𝑦)
1716a1i 9 . . . . . . . 8 (𝜑 → (𝑦𝑅𝑧𝑧𝑅𝑦))
1817rexbidv 2491 . . . . . . 7 (𝜑 → (∃𝑧𝐵 𝑦𝑅𝑧 ↔ ∃𝑧𝐵 𝑧𝑅𝑦))
1914, 18imbi12d 234 . . . . . 6 (𝜑 → ((𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
2019ralbidv 2490 . . . . 5 (𝜑 → (∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
2112, 20anbi12d 473 . . . 4 (𝜑 → ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ↔ (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦))))
2221riotabidv 5854 . . 3 (𝜑 → (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))) = (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦))))
236, 22eqtrd 2222 . 2 (𝜑 → sup(𝐵, 𝐴, 𝑅) = (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦))))
241, 23eqtrid 2234 1 (𝜑 → inf(𝐵, 𝐴, 𝑅) = (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160  wral 2468  wrex 2469   class class class wbr 4018  ccnv 4643  crio 5851  supcsup 7012  infcinf 7013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-cnv 4652  df-iota 5196  df-riota 5852  df-sup 7014  df-inf 7015
This theorem is referenced by: (None)
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