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Theorem supiso 8941
 Description: Image of a supremum under an isomorphism. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypotheses
Ref Expression
supiso.1 (𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
supiso.2 (𝜑𝐶𝐴)
supisoex.3 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))
supiso.4 (𝜑𝑅 Or 𝐴)
Assertion
Ref Expression
supiso (𝜑 → sup((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐶,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem supiso
Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supiso.4 . . 3 (𝜑𝑅 Or 𝐴)
2 supiso.1 . . . 4 (𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))
3 isoso 7090 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Or 𝐴𝑆 Or 𝐵))
42, 3syl 17 . . 3 (𝜑 → (𝑅 Or 𝐴𝑆 Or 𝐵))
51, 4mpbid 235 . 2 (𝜑𝑆 Or 𝐵)
6 isof1o 7065 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
7 f1of 6599 . . . 4 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
82, 6, 73syl 18 . . 3 (𝜑𝐹:𝐴𝐵)
9 supisoex.3 . . . 4 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))
101, 9supcl 8924 . . 3 (𝜑 → sup(𝐶, 𝐴, 𝑅) ∈ 𝐴)
118, 10ffvelrnd 6839 . 2 (𝜑 → (𝐹‘sup(𝐶, 𝐴, 𝑅)) ∈ 𝐵)
121, 9supub 8925 . . . . . 6 (𝜑 → (𝑢𝐶 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑢))
1312ralrimiv 3148 . . . . 5 (𝜑 → ∀𝑢𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑢)
141, 9suplub 8926 . . . . . . 7 (𝜑 → ((𝑢𝐴𝑢𝑅sup(𝐶, 𝐴, 𝑅)) → ∃𝑧𝐶 𝑢𝑅𝑧))
1514expd 419 . . . . . 6 (𝜑 → (𝑢𝐴 → (𝑢𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧𝐶 𝑢𝑅𝑧)))
1615ralrimiv 3148 . . . . 5 (𝜑 → ∀𝑢𝐴 (𝑢𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧𝐶 𝑢𝑅𝑧))
17 supiso.2 . . . . . . 7 (𝜑𝐶𝐴)
182, 17supisolem 8939 . . . . . 6 ((𝜑 ∧ sup(𝐶, 𝐴, 𝑅) ∈ 𝐴) → ((∀𝑢𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑢 ∧ ∀𝑢𝐴 (𝑢𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧𝐶 𝑢𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))))
1910, 18mpdan 686 . . . . 5 (𝜑 → ((∀𝑢𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑢 ∧ ∀𝑢𝐴 (𝑢𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧𝐶 𝑢𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))))
2013, 16, 19mpbi2and 711 . . . 4 (𝜑 → (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
2120simpld 498 . . 3 (𝜑 → ∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
2221r19.21bi 3173 . 2 ((𝜑𝑤 ∈ (𝐹𝐶)) → ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
2320simprd 499 . . . 4 (𝜑 → ∀𝑤𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))
2423r19.21bi 3173 . . 3 ((𝜑𝑤𝐵) → (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))
2524impr 458 . 2 ((𝜑 ∧ (𝑤𝐵𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)))) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)
265, 11, 22, 25eqsupd 8923 1 (𝜑 → sup((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ⊆ wss 3883   class class class wbr 5034   Or wor 5441   “ cima 5526  ⟶wf 6328  –1-1-onto→wf1o 6331  ‘cfv 6332   Isom wiso 6333  supcsup 8906 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-po 5442  df-so 5443  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-isom 6341  df-riota 7103  df-sup 8908 This theorem is referenced by:  infiso  8974  infrenegsup  11629
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