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Theorem eqsupti 6941
Description: Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.)
Hypothesis
Ref Expression
supmoti.ti ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
Assertion
Ref Expression
eqsupti (𝜑 → ((𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧)) → sup(𝐵, 𝐴, 𝑅) = 𝐶))
Distinct variable groups:   𝑢,𝐴,𝑣,𝑦,𝑧   𝑦,𝐵,𝑧   𝑢,𝑅,𝑣,𝑦,𝑧   𝜑,𝑢,𝑣   𝑦,𝑢,𝑣,𝐶   𝑢,𝐵,𝑣,𝑧
Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐶(𝑧)

Proof of Theorem eqsupti
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 supmoti.ti . . . . 5 ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
21adantlr 469 . . . 4 (((𝜑 ∧ (𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧))) ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
3 breq1 3969 . . . . . . . . . 10 (𝑥 = 𝐶 → (𝑥𝑅𝑦𝐶𝑅𝑦))
43notbid 657 . . . . . . . . 9 (𝑥 = 𝐶 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝐶𝑅𝑦))
54ralbidv 2457 . . . . . . . 8 (𝑥 = 𝐶 → (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ↔ ∀𝑦𝐵 ¬ 𝐶𝑅𝑦))
6 breq2 3970 . . . . . . . . . 10 (𝑥 = 𝐶 → (𝑦𝑅𝑥𝑦𝑅𝐶))
76imbi1d 230 . . . . . . . . 9 (𝑥 = 𝐶 → ((𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧)))
87ralbidv 2457 . . . . . . . 8 (𝑥 = 𝐶 → (∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧) ↔ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧)))
95, 8anbi12d 465 . . . . . . 7 (𝑥 = 𝐶 → ((∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) ↔ (∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧))))
109rspcev 2816 . . . . . 6 ((𝐶𝐴 ∧ (∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧))) → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
11103impb 1181 . . . . 5 ((𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧)) → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
1211adantl 275 . . . 4 ((𝜑 ∧ (𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧))) → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
132, 12supval2ti 6940 . . 3 ((𝜑 ∧ (𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧))) → sup(𝐵, 𝐴, 𝑅) = (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))))
14 3simpc 981 . . . . 5 ((𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧)) → (∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧)))
1514adantl 275 . . . 4 ((𝜑 ∧ (𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧))) → (∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧)))
16 simpr1 988 . . . . 5 ((𝜑 ∧ (𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧))) → 𝐶𝐴)
172, 12supeuti 6939 . . . . 5 ((𝜑 ∧ (𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧))) → ∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
189riota2 5803 . . . . 5 ((𝐶𝐴 ∧ ∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))) → ((∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧)) ↔ (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))) = 𝐶))
1916, 17, 18syl2anc 409 . . . 4 ((𝜑 ∧ (𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧))) → ((∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧)) ↔ (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))) = 𝐶))
2015, 19mpbid 146 . . 3 ((𝜑 ∧ (𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧))) → (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))) = 𝐶)
2113, 20eqtrd 2190 . 2 ((𝜑 ∧ (𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧))) → sup(𝐵, 𝐴, 𝑅) = 𝐶)
2221ex 114 1 (𝜑 → ((𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧)) → sup(𝐵, 𝐴, 𝑅) = 𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  w3a 963   = wceq 1335  wcel 2128  wral 2435  wrex 2436  ∃!wreu 2437   class class class wbr 3966  crio 5780  supcsup 6927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-iota 5136  df-riota 5781  df-sup 6929
This theorem is referenced by:  eqsuptid  6942  eqinfti  6965  maxabs  11113  xrmaxif  11152  bezoutlemsup  11897  suplociccex  13045
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