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Theorem fisupclrnmpt 45348
Description: A nonempty finite indexed set contains its supremum. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
fisupclrnmpt.x 𝑥𝜑
fisupclrnmpt.r (𝜑𝑅 Or 𝐴)
fisupclrnmpt.b (𝜑𝐵 ∈ Fin)
fisupclrnmpt.n (𝜑𝐵 ≠ ∅)
fisupclrnmpt.c ((𝜑𝑥𝐵) → 𝐶𝐴)
Assertion
Ref Expression
fisupclrnmpt (𝜑 → sup(ran (𝑥𝐵𝐶), 𝐴, 𝑅) ∈ 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)   𝑅(𝑥)

Proof of Theorem fisupclrnmpt
StepHypRef Expression
1 fisupclrnmpt.x . . 3 𝑥𝜑
2 eqid 2735 . . 3 (𝑥𝐵𝐶) = (𝑥𝐵𝐶)
3 fisupclrnmpt.c . . 3 ((𝜑𝑥𝐵) → 𝐶𝐴)
41, 2, 3rnmptssd 45139 . 2 (𝜑 → ran (𝑥𝐵𝐶) ⊆ 𝐴)
5 fisupclrnmpt.r . . 3 (𝜑𝑅 Or 𝐴)
6 fisupclrnmpt.b . . . 4 (𝜑𝐵 ∈ Fin)
72rnmptfi 45114 . . . 4 (𝐵 ∈ Fin → ran (𝑥𝐵𝐶) ∈ Fin)
86, 7syl 17 . . 3 (𝜑 → ran (𝑥𝐵𝐶) ∈ Fin)
9 fisupclrnmpt.n . . . 4 (𝜑𝐵 ≠ ∅)
101, 3, 2, 9rnmptn0 6266 . . 3 (𝜑 → ran (𝑥𝐵𝐶) ≠ ∅)
11 fisupcl 9507 . . 3 ((𝑅 Or 𝐴 ∧ (ran (𝑥𝐵𝐶) ∈ Fin ∧ ran (𝑥𝐵𝐶) ≠ ∅ ∧ ran (𝑥𝐵𝐶) ⊆ 𝐴)) → sup(ran (𝑥𝐵𝐶), 𝐴, 𝑅) ∈ ran (𝑥𝐵𝐶))
125, 8, 10, 4, 11syl13anc 1371 . 2 (𝜑 → sup(ran (𝑥𝐵𝐶), 𝐴, 𝑅) ∈ ran (𝑥𝐵𝐶))
134, 12sseldd 3996 1 (𝜑 → sup(ran (𝑥𝐵𝐶), 𝐴, 𝑅) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wnf 1780  wcel 2106  wne 2938  wss 3963  c0 4339  cmpt 5231   Or wor 5596  ran crn 5690  Fincfn 8984  supcsup 9478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-en 8985  df-dom 8986  df-fin 8988  df-sup 9480
This theorem is referenced by:  uzublem  45380  limsupubuzlem  45668
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