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Theorem infiso 8948
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 7058 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐹 Isom 𝑅, 𝑆(𝐴, 𝐵))
31, 2sylib 221 . . 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 8927 . . 3 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))
8 cnvso 6112 . . . 4 (𝑅 Or 𝐴𝑅 Or 𝐴)
95, 8sylib 221 . . 3 (𝜑𝑅 Or 𝐴)
103, 4, 7, 9supiso 8915 . 2 (𝜑 → sup((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))
11 df-inf 8883 . 2 inf((𝐹𝐶), 𝐵, 𝑆) = sup((𝐹𝐶), 𝐵, 𝑆)
12 df-inf 8883 . . 3 inf(𝐶, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)
1312fveq2i 6646 . 2 (𝐹‘inf(𝐶, 𝐴, 𝑅)) = (𝐹‘sup(𝐶, 𝐴, 𝑅))
1410, 11, 133eqtr4g 2881 1 (𝜑 → inf((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘inf(𝐶, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wral 3126  wrex 3127  wss 3910   class class class wbr 5039   Or wor 5446  ccnv 5527  cima 5531  cfv 6328   Isom wiso 6329  supcsup 8880  infcinf 8881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-po 5447  df-so 5448  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7088  df-sup 8882  df-inf 8883
This theorem is referenced by: (None)
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