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

Theorem infiso 8373
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 6546 . . . 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 8353 . . 3 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))
8 cnvso 5643 . . . 4 (𝑅 Or 𝐴𝑅 Or 𝐴)
95, 8sylib 208 . . 3 (𝜑𝑅 Or 𝐴)
103, 4, 7, 9supiso 8341 . 2 (𝜑 → sup((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))
11 df-inf 8309 . 2 inf((𝐹𝐶), 𝐵, 𝑆) = sup((𝐹𝐶), 𝐵, 𝑆)
12 df-inf 8309 . . 3 inf(𝐶, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)
1312fveq2i 6161 . 2 (𝐹‘inf(𝐶, 𝐴, 𝑅)) = (𝐹‘sup(𝐶, 𝐴, 𝑅))
1410, 11, 133eqtr4g 2680 1 (𝜑 → inf((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘inf(𝐶, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wral 2908  wrex 2909  wss 3560   class class class wbr 4623   Or wor 5004  ccnv 5083  cima 5087  cfv 5857   Isom wiso 5858  supcsup 8306  infcinf 8307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-po 5005  df-so 5006  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-sup 8308  df-inf 8309
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator