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Theorem infiso 9548
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 7351 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐹 Isom 𝑅, 𝑆(𝐴, 𝐵))
31, 2sylib 218 . . 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 9527 . . 3 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))
8 cnvso 6308 . . . 4 (𝑅 Or 𝐴𝑅 Or 𝐴)
95, 8sylib 218 . . 3 (𝜑𝑅 Or 𝐴)
103, 4, 7, 9supiso 9515 . 2 (𝜑 → sup((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))
11 df-inf 9483 . 2 inf((𝐹𝐶), 𝐵, 𝑆) = sup((𝐹𝐶), 𝐵, 𝑆)
12 df-inf 9483 . . 3 inf(𝐶, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)
1312fveq2i 6909 . 2 (𝐹‘inf(𝐶, 𝐴, 𝑅)) = (𝐹‘sup(𝐶, 𝐴, 𝑅))
1410, 11, 133eqtr4g 2802 1 (𝜑 → inf((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘inf(𝐶, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wral 3061  wrex 3070  wss 3951   class class class wbr 5143   Or wor 5591  ccnv 5684  cima 5688  cfv 6561   Isom wiso 6562  supcsup 9480  infcinf 9481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-sup 9482  df-inf 9483
This theorem is referenced by: (None)
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