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

Theorem infiso 8566
Description: Image of an infimum under an isomorphism. (Contributed by AV, 4-Sep-2020.)
Hypotheses
Ref Expression
infiso.1 (𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
infiso.2 (𝜑𝐶𝐴)
infiso.3 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐶 𝑧𝑅𝑦)))
infiso.4 (𝜑𝑅 Or 𝐴)
Assertion
Ref Expression
infiso (𝜑 → inf((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘inf(𝐶, 𝐴, 𝑅)))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

Proof of Theorem infiso
StepHypRef Expression
1 infiso.1 . . . 4 (𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
2 isocnv2 6732 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐹 Isom 𝑅, 𝑆(𝐴, 𝐵))
31, 2sylib 208 . . 3 (𝜑𝐹 Isom 𝑅, 𝑆(𝐴, 𝐵))
4 infiso.2 . . 3 (𝜑𝐶𝐴)
5 infiso.4 . . . 4 (𝜑𝑅 Or 𝐴)
6 infiso.3 . . . 4 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐶 𝑧𝑅𝑦)))
75, 6infcllem 8546 . . 3 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))
8 cnvso 5823 . . . 4 (𝑅 Or 𝐴𝑅 Or 𝐴)
95, 8sylib 208 . . 3 (𝜑𝑅 Or 𝐴)
103, 4, 7, 9supiso 8534 . 2 (𝜑 → sup((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))
11 df-inf 8502 . 2 inf((𝐹𝐶), 𝐵, 𝑆) = sup((𝐹𝐶), 𝐵, 𝑆)
12 df-inf 8502 . . 3 inf(𝐶, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)
1312fveq2i 6343 . 2 (𝐹‘inf(𝐶, 𝐴, 𝑅)) = (𝐹‘sup(𝐶, 𝐴, 𝑅))
1410, 11, 133eqtr4g 2807 1 (𝜑 → inf((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘inf(𝐶, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1620  wral 3038  wrex 3039  wss 3703   class class class wbr 4792   Or wor 5174  ccnv 5253  cima 5257  cfv 6037   Isom wiso 6038  supcsup 8499  infcinf 8500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rmo 3046  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-po 5175  df-so 5176  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-isom 6046  df-riota 6762  df-sup 8501  df-inf 8502
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator