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Theorem supisoti 6610
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 2449 . . . . . 6 (𝜑 → ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
3 supiso.1 . . . . . . 7 (𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
4 isoti 6607 . . . . . . 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 145 . . . . 5 (𝜑 → ∀𝑢𝐵𝑣𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
76r19.21bi 2455 . . . 4 ((𝜑𝑢𝐵) → ∀𝑣𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
87r19.21bi 2455 . . 3 (((𝜑𝑢𝐵) ∧ 𝑣𝐵) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
98anasss 391 . 2 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
10 isof1o 5524 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
11 f1of 5199 . . . 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 6598 . . 3 (𝜑 → sup(𝐶, 𝐴, 𝑅) ∈ 𝐴)
1512, 14ffvelrnd 5378 . 2 (𝜑 → (𝐹‘sup(𝐶, 𝐴, 𝑅)) ∈ 𝐵)
161, 13supubti 6599 . . . . . 6 (𝜑 → (𝑗𝐶 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑗))
1716ralrimiv 2439 . . . . 5 (𝜑 → ∀𝑗𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑗)
181, 13suplubti 6600 . . . . . . 7 (𝜑 → ((𝑗𝐴𝑗𝑅sup(𝐶, 𝐴, 𝑅)) → ∃𝑧𝐶 𝑗𝑅𝑧))
1918expd 254 . . . . . 6 (𝜑 → (𝑗𝐴 → (𝑗𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧𝐶 𝑗𝑅𝑧)))
2019ralrimiv 2439 . . . . 5 (𝜑 → ∀𝑗𝐴 (𝑗𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧𝐶 𝑗𝑅𝑧))
21 supiso.2 . . . . . . 7 (𝜑𝐶𝐴)
223, 21supisolem 6608 . . . . . 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 412 . . . . 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 885 . . . 4 (𝜑 → (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑘 ∈ (𝐹𝐶)𝑤𝑆𝑘)))
2524simpld 110 . . 3 (𝜑 → ∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
2625r19.21bi 2455 . 2 ((𝜑𝑤 ∈ (𝐹𝐶)) → ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
2724simprd 112 . . . 4 (𝜑 → ∀𝑤𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑘 ∈ (𝐹𝐶)𝑤𝑆𝑘))
2827r19.21bi 2455 . . 3 ((𝜑𝑤𝐵) → (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑘 ∈ (𝐹𝐶)𝑤𝑆𝑘))
2928impr 371 . 2 ((𝜑 ∧ (𝑤𝐵𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)))) → ∃𝑘 ∈ (𝐹𝐶)𝑤𝑆𝑘)
309, 15, 26, 29eqsuptid 6597 1 (𝜑 → sup((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  wral 2353  wrex 2354  wss 2984   class class class wbr 3811  cima 4402  wf 4963  1-1-ontowf1o 4966  cfv 4967   Isom wiso 4968  supcsup 6582
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4083  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-isom 4976  df-riota 5545  df-sup 6584
This theorem is referenced by:  infisoti  6632  infrenegsupex  8975
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