MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  supiso Structured version   Visualization version   GIF version

Theorem supiso 8422
Description: Image of a supremum under an isomorphism. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypotheses
Ref Expression
supiso.1 (𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
supiso.2 (𝜑𝐶𝐴)
supisoex.3 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))
supiso.4 (𝜑𝑅 Or 𝐴)
Assertion
Ref Expression
supiso (𝜑 → sup((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐶,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem supiso
Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supiso.4 . . 3 (𝜑𝑅 Or 𝐴)
2 supiso.1 . . . 4 (𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
3 isoso 6638 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Or 𝐴𝑆 Or 𝐵))
42, 3syl 17 . . 3 (𝜑 → (𝑅 Or 𝐴𝑆 Or 𝐵))
51, 4mpbid 222 . 2 (𝜑𝑆 Or 𝐵)
6 isof1o 6613 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
7 f1of 6175 . . . 4 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
82, 6, 73syl 18 . . 3 (𝜑𝐹:𝐴𝐵)
9 supisoex.3 . . . 4 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))
101, 9supcl 8405 . . 3 (𝜑 → sup(𝐶, 𝐴, 𝑅) ∈ 𝐴)
118, 10ffvelrnd 6400 . 2 (𝜑 → (𝐹‘sup(𝐶, 𝐴, 𝑅)) ∈ 𝐵)
121, 9supub 8406 . . . . . 6 (𝜑 → (𝑢𝐶 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑢))
1312ralrimiv 2994 . . . . 5 (𝜑 → ∀𝑢𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑢)
141, 9suplub 8407 . . . . . . 7 (𝜑 → ((𝑢𝐴𝑢𝑅sup(𝐶, 𝐴, 𝑅)) → ∃𝑧𝐶 𝑢𝑅𝑧))
1514expd 451 . . . . . 6 (𝜑 → (𝑢𝐴 → (𝑢𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧𝐶 𝑢𝑅𝑧)))
1615ralrimiv 2994 . . . . 5 (𝜑 → ∀𝑢𝐴 (𝑢𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧𝐶 𝑢𝑅𝑧))
17 supiso.2 . . . . . . 7 (𝜑𝐶𝐴)
182, 17supisolem 8420 . . . . . 6 ((𝜑 ∧ sup(𝐶, 𝐴, 𝑅) ∈ 𝐴) → ((∀𝑢𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑢 ∧ ∀𝑢𝐴 (𝑢𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧𝐶 𝑢𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))))
1910, 18mpdan 703 . . . . 5 (𝜑 → ((∀𝑢𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑢 ∧ ∀𝑢𝐴 (𝑢𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧𝐶 𝑢𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))))
2013, 16, 19mpbi2and 976 . . . 4 (𝜑 → (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
2120simpld 474 . . 3 (𝜑 → ∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
2221r19.21bi 2961 . 2 ((𝜑𝑤 ∈ (𝐹𝐶)) → ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
2320simprd 478 . . . 4 (𝜑 → ∀𝑤𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))
2423r19.21bi 2961 . . 3 ((𝜑𝑤𝐵) → (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))
2524impr 648 . 2 ((𝜑 ∧ (𝑤𝐵𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)))) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)
265, 11, 22, 25eqsupd 8404 1 (𝜑 → sup((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  wss 3607   class class class wbr 4685   Or wor 5063  cima 5146  wf 5922  1-1-ontowf1o 5925  cfv 5926   Isom wiso 5927  supcsup 8387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-sup 8389
This theorem is referenced by:  infiso  8454  infrenegsup  11044
  Copyright terms: Public domain W3C validator