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Theorem supiso 8326
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 6553 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Or 𝐴𝑆 Or 𝐵))
42, 3syl 17 . . 3 (𝜑 → (𝑅 Or 𝐴𝑆 Or 𝐵))
51, 4mpbid 222 . 2 (𝜑𝑆 Or 𝐵)
6 isof1o 6528 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
7 f1of 6096 . . . 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 8309 . . 3 (𝜑 → sup(𝐶, 𝐴, 𝑅) ∈ 𝐴)
118, 10ffvelrnd 6317 . 2 (𝜑 → (𝐹‘sup(𝐶, 𝐴, 𝑅)) ∈ 𝐵)
121, 9supub 8310 . . . . . 6 (𝜑 → (𝑢𝐶 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑢))
1312ralrimiv 2964 . . . . 5 (𝜑 → ∀𝑢𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑢)
141, 9suplub 8311 . . . . . . 7 (𝜑 → ((𝑢𝐴𝑢𝑅sup(𝐶, 𝐴, 𝑅)) → ∃𝑧𝐶 𝑢𝑅𝑧))
1514expd 452 . . . . . 6 (𝜑 → (𝑢𝐴 → (𝑢𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧𝐶 𝑢𝑅𝑧)))
1615ralrimiv 2964 . . . . 5 (𝜑 → ∀𝑢𝐴 (𝑢𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧𝐶 𝑢𝑅𝑧))
17 supiso.2 . . . . . . 7 (𝜑𝐶𝐴)
182, 17supisolem 8324 . . . . . 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 701 . . . . 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 955 . . . 4 (𝜑 → (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
2120simpld 475 . . 3 (𝜑 → ∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
2221r19.21bi 2932 . 2 ((𝜑𝑤 ∈ (𝐹𝐶)) → ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
2320simprd 479 . . . 4 (𝜑 → ∀𝑤𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))
2423r19.21bi 2932 . . 3 ((𝜑𝑤𝐵) → (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))
2524impr 648 . 2 ((𝜑 ∧ (𝑤𝐵𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)))) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)
265, 11, 22, 25eqsupd 8308 1 (𝜑 → sup((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wral 2912  wrex 2913  wss 3560   class class class wbr 4618   Or wor 4999  cima 5082  wf 5846  1-1-ontowf1o 5849  cfv 5850   Isom wiso 5851  supcsup 8291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-po 5000  df-so 5001  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-sup 8293
This theorem is referenced by:  infiso  8358  infrenegsup  10951
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