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Theorem supisoti 6414
Description: Image of a supremum under an isomorphism. (Contributed by Jim Kingdon, 26-Nov-2021.)
Hypotheses
Ref Expression
supiso.1 (𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
supiso.2 (𝜑𝐶𝐴)
supisoex.3 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))
supisoti.ti ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
Assertion
Ref Expression
supisoti (𝜑 → sup((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))
Distinct variable groups:   𝑣,𝑢,𝑥,𝑦,𝑧,𝐴   𝑢,𝐶,𝑣,𝑥,𝑦,𝑧   𝜑,𝑢   𝑢,𝐹,𝑣,𝑥,𝑦,𝑧   𝑢,𝑅,𝑥,𝑦,𝑧   𝑢,𝑆,𝑣,𝑥,𝑦,𝑧   𝑢,𝐵,𝑣,𝑥,𝑦,𝑧   𝑣,𝑅   𝜑,𝑣,𝑥
Allowed substitution hints:   𝜑(𝑦,𝑧)

Proof of Theorem supisoti
Dummy variables 𝑤 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supisoti.ti . . . . . . 7 ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
21ralrimivva 2418 . . . . . 6 (𝜑 → ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
3 supiso.1 . . . . . . 7 (𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
4 isoti 6411 . . . . . . 7 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ ∀𝑢𝐵𝑣𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢))))
53, 4syl 14 . . . . . 6 (𝜑 → (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ ∀𝑢𝐵𝑣𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢))))
62, 5mpbid 139 . . . . 5 (𝜑 → ∀𝑢𝐵𝑣𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
76r19.21bi 2424 . . . 4 ((𝜑𝑢𝐵) → ∀𝑣𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
87r19.21bi 2424 . . 3 (((𝜑𝑢𝐵) ∧ 𝑣𝐵) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
98anasss 385 . 2 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
10 isof1o 5475 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
11 f1of 5154 . . . 4 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
123, 10, 113syl 17 . . 3 (𝜑𝐹:𝐴𝐵)
13 supisoex.3 . . . 4 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))
141, 13supclti 6404 . . 3 (𝜑 → sup(𝐶, 𝐴, 𝑅) ∈ 𝐴)
1512, 14ffvelrnd 5331 . 2 (𝜑 → (𝐹‘sup(𝐶, 𝐴, 𝑅)) ∈ 𝐵)
161, 13supubti 6405 . . . . . 6 (𝜑 → (𝑗𝐶 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑗))
1716ralrimiv 2408 . . . . 5 (𝜑 → ∀𝑗𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑗)
181, 13suplubti 6406 . . . . . . 7 (𝜑 → ((𝑗𝐴𝑗𝑅sup(𝐶, 𝐴, 𝑅)) → ∃𝑧𝐶 𝑗𝑅𝑧))
1918expd 249 . . . . . 6 (𝜑 → (𝑗𝐴 → (𝑗𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧𝐶 𝑗𝑅𝑧)))
2019ralrimiv 2408 . . . . 5 (𝜑 → ∀𝑗𝐴 (𝑗𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧𝐶 𝑗𝑅𝑧))
21 supiso.2 . . . . . . 7 (𝜑𝐶𝐴)
223, 21supisolem 6412 . . . . . 6 ((𝜑 ∧ sup(𝐶, 𝐴, 𝑅) ∈ 𝐴) → ((∀𝑗𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑗 ∧ ∀𝑗𝐴 (𝑗𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧𝐶 𝑗𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑘 ∈ (𝐹𝐶)𝑤𝑆𝑘))))
2314, 22mpdan 406 . . . . 5 (𝜑 → ((∀𝑗𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑗 ∧ ∀𝑗𝐴 (𝑗𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧𝐶 𝑗𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑘 ∈ (𝐹𝐶)𝑤𝑆𝑘))))
2417, 20, 23mpbi2and 861 . . . 4 (𝜑 → (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑘 ∈ (𝐹𝐶)𝑤𝑆𝑘)))
2524simpld 109 . . 3 (𝜑 → ∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
2625r19.21bi 2424 . 2 ((𝜑𝑤 ∈ (𝐹𝐶)) → ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
2724simprd 111 . . . 4 (𝜑 → ∀𝑤𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑘 ∈ (𝐹𝐶)𝑤𝑆𝑘))
2827r19.21bi 2424 . . 3 ((𝜑𝑤𝐵) → (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑘 ∈ (𝐹𝐶)𝑤𝑆𝑘))
2928impr 365 . 2 ((𝜑 ∧ (𝑤𝐵𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)))) → ∃𝑘 ∈ (𝐹𝐶)𝑤𝑆𝑘)
309, 15, 26, 29eqsuptid 6403 1 (𝜑 → sup((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  wral 2323  wrex 2324  wss 2945   class class class wbr 3792  cima 4376  wf 4926  1-1-ontowf1o 4929  cfv 4930   Isom wiso 4931  supcsup 6388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-rab 2332  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-isom 4939  df-riota 5496  df-sup 6390
This theorem is referenced by: (None)
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