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Theorem supiso 8264
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 6498 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Or 𝐴𝑆 Or 𝐵))
42, 3syl 17 . . 3 (𝜑 → (𝑅 Or 𝐴𝑆 Or 𝐵))
51, 4mpbid 221 . 2 (𝜑𝑆 Or 𝐵)
6 isof1o 6473 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
7 f1of 6050 . . . 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 8247 . . 3 (𝜑 → sup(𝐶, 𝐴, 𝑅) ∈ 𝐴)
118, 10ffvelrnd 6268 . 2 (𝜑 → (𝐹‘sup(𝐶, 𝐴, 𝑅)) ∈ 𝐵)
121, 9supub 8248 . . . . . 6 (𝜑 → (𝑢𝐶 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑢))
1312ralrimiv 2948 . . . . 5 (𝜑 → ∀𝑢𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑢)
141, 9suplub 8249 . . . . . . 7 (𝜑 → ((𝑢𝐴𝑢𝑅sup(𝐶, 𝐴, 𝑅)) → ∃𝑧𝐶 𝑢𝑅𝑧))
1514expd 451 . . . . . 6 (𝜑 → (𝑢𝐴 → (𝑢𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧𝐶 𝑢𝑅𝑧)))
1615ralrimiv 2948 . . . . 5 (𝜑 → ∀𝑢𝐴 (𝑢𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧𝐶 𝑢𝑅𝑧))
17 supiso.2 . . . . . . 7 (𝜑𝐶𝐴)
182, 17supisolem 8262 . . . . . 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 699 . . . . 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 958 . . . 4 (𝜑 → (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
2120simpld 474 . . 3 (𝜑 → ∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
2221r19.21bi 2916 . 2 ((𝜑𝑤 ∈ (𝐹𝐶)) → ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
2320simprd 478 . . . 4 (𝜑 → ∀𝑤𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))
2423r19.21bi 2916 . . 3 ((𝜑𝑤𝐵) → (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))
2524impr 647 . 2 ((𝜑 ∧ (𝑤𝐵𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)))) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)
265, 11, 22, 25eqsupd 8246 1 (𝜑 → sup((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  wrex 2897  wss 3540   class class class wbr 4583   Or wor 4958  cima 5041  wf 5800  1-1-ontowf1o 5803  cfv 5804   Isom wiso 5805  supcsup 8229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-sup 8231
This theorem is referenced by:  infiso  8296  infrenegsup  10883
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