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Theorem supisoti 7026
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 2571 . . . . . 6 (𝜑 → ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
3 supiso.1 . . . . . . 7 (𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
4 isoti 7023 . . . . . . 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 147 . . . . 5 (𝜑 → ∀𝑢𝐵𝑣𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
76r19.21bi 2577 . . . 4 ((𝜑𝑢𝐵) → ∀𝑣𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
87r19.21bi 2577 . . 3 (((𝜑𝑢𝐵) ∧ 𝑣𝐵) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
98anasss 399 . 2 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢)))
10 isof1o 5823 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
11 f1of 5475 . . . 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 7014 . . 3 (𝜑 → sup(𝐶, 𝐴, 𝑅) ∈ 𝐴)
1512, 14ffvelcdmd 5667 . 2 (𝜑 → (𝐹‘sup(𝐶, 𝐴, 𝑅)) ∈ 𝐵)
161, 13supubti 7015 . . . . . 6 (𝜑 → (𝑗𝐶 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑗))
1716ralrimiv 2561 . . . . 5 (𝜑 → ∀𝑗𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑗)
181, 13suplubti 7016 . . . . . . 7 (𝜑 → ((𝑗𝐴𝑗𝑅sup(𝐶, 𝐴, 𝑅)) → ∃𝑧𝐶 𝑗𝑅𝑧))
1918expd 258 . . . . . 6 (𝜑 → (𝑗𝐴 → (𝑗𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧𝐶 𝑗𝑅𝑧)))
2019ralrimiv 2561 . . . . 5 (𝜑 → ∀𝑗𝐴 (𝑗𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧𝐶 𝑗𝑅𝑧))
21 supiso.2 . . . . . . 7 (𝜑𝐶𝐴)
223, 21supisolem 7024 . . . . . 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 421 . . . . 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 944 . . . 4 (𝜑 → (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑘 ∈ (𝐹𝐶)𝑤𝑆𝑘)))
2524simpld 112 . . 3 (𝜑 → ∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
2625r19.21bi 2577 . 2 ((𝜑𝑤 ∈ (𝐹𝐶)) → ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
2724simprd 114 . . . 4 (𝜑 → ∀𝑤𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑘 ∈ (𝐹𝐶)𝑤𝑆𝑘))
2827r19.21bi 2577 . . 3 ((𝜑𝑤𝐵) → (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑘 ∈ (𝐹𝐶)𝑤𝑆𝑘))
2928impr 379 . 2 ((𝜑 ∧ (𝑤𝐵𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)))) → ∃𝑘 ∈ (𝐹𝐶)𝑤𝑆𝑘)
309, 15, 26, 29eqsuptid 7013 1 (𝜑 → sup((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105   = wceq 1363  wcel 2159  wral 2467  wrex 2468  wss 3143   class class class wbr 4017  cima 4643  wf 5226  1-1-ontowf1o 5229  cfv 5230   Isom wiso 5231  supcsup 6998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-br 4018  df-opab 4079  df-mpt 4080  df-id 4307  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-isom 5239  df-riota 5846  df-sup 7000
This theorem is referenced by:  infisoti  7048  infrenegsupex  9611  infxrnegsupex  11288
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