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Theorem supiso 8546
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 6761 . . . 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 6736 . . . 4 (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐹:𝐴1-1-onto𝐵)
7 f1of 6298 . . . 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 8529 . . 3 (𝜑 → sup(𝐶, 𝐴, 𝑅) ∈ 𝐴)
118, 10ffvelrnd 6523 . 2 (𝜑 → (𝐹‘sup(𝐶, 𝐴, 𝑅)) ∈ 𝐵)
121, 9supub 8530 . . . . . 6 (𝜑 → (𝑢𝐶 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑢))
1312ralrimiv 3103 . . . . 5 (𝜑 → ∀𝑢𝐶 ¬ sup(𝐶, 𝐴, 𝑅)𝑅𝑢)
141, 9suplub 8531 . . . . . . 7 (𝜑 → ((𝑢𝐴𝑢𝑅sup(𝐶, 𝐴, 𝑅)) → ∃𝑧𝐶 𝑢𝑅𝑧))
1514expd 451 . . . . . 6 (𝜑 → (𝑢𝐴 → (𝑢𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧𝐶 𝑢𝑅𝑧)))
1615ralrimiv 3103 . . . . 5 (𝜑 → ∀𝑢𝐴 (𝑢𝑅sup(𝐶, 𝐴, 𝑅) → ∃𝑧𝐶 𝑢𝑅𝑧))
17 supiso.2 . . . . . . 7 (𝜑𝐶𝐴)
182, 17supisolem 8544 . . . . . 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 705 . . . . 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 994 . . . 4 (𝜑 → (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
2120simpld 477 . . 3 (𝜑 → ∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
2221r19.21bi 3070 . 2 ((𝜑𝑤 ∈ (𝐹𝐶)) → ¬ (𝐹‘sup(𝐶, 𝐴, 𝑅))𝑆𝑤)
2320simprd 482 . . . 4 (𝜑 → ∀𝑤𝐵 (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))
2423r19.21bi 3070 . . 3 ((𝜑𝑤𝐵) → (𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))
2524impr 650 . 2 ((𝜑 ∧ (𝑤𝐵𝑤𝑆(𝐹‘sup(𝐶, 𝐴, 𝑅)))) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)
265, 11, 22, 25eqsupd 8528 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 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  wss 3715   class class class wbr 4804   Or wor 5186  cima 5269  wf 6045  1-1-ontowf1o 6048  cfv 6049   Isom wiso 6050  supcsup 8511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-sup 8513
This theorem is referenced by:  infiso  8578  infrenegsup  11198
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