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Theorem fisupclrnmpt 45409
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 2737 . . 3 (𝑥𝐵𝐶) = (𝑥𝐵𝐶)
3 fisupclrnmpt.c . . 3 ((𝜑𝑥𝐵) → 𝐶𝐴)
41, 2, 3rnmptssd 45201 . 2 (𝜑 → ran (𝑥𝐵𝐶) ⊆ 𝐴)
5 fisupclrnmpt.r . . 3 (𝜑𝑅 Or 𝐴)
6 fisupclrnmpt.b . . . 4 (𝜑𝐵 ∈ Fin)
72rnmptfi 45176 . . . 4 (𝐵 ∈ Fin → ran (𝑥𝐵𝐶) ∈ Fin)
86, 7syl 17 . . 3 (𝜑 → ran (𝑥𝐵𝐶) ∈ Fin)
9 fisupclrnmpt.n . . . 4 (𝜑𝐵 ≠ ∅)
101, 3, 2, 9rnmptn0 6264 . . 3 (𝜑 → ran (𝑥𝐵𝐶) ≠ ∅)
11 fisupcl 9509 . . 3 ((𝑅 Or 𝐴 ∧ (ran (𝑥𝐵𝐶) ∈ Fin ∧ ran (𝑥𝐵𝐶) ≠ ∅ ∧ ran (𝑥𝐵𝐶) ⊆ 𝐴)) → sup(ran (𝑥𝐵𝐶), 𝐴, 𝑅) ∈ ran (𝑥𝐵𝐶))
125, 8, 10, 4, 11syl13anc 1374 . 2 (𝜑 → sup(ran (𝑥𝐵𝐶), 𝐴, 𝑅) ∈ ran (𝑥𝐵𝐶))
134, 12sseldd 3984 1 (𝜑 → sup(ran (𝑥𝐵𝐶), 𝐴, 𝑅) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wnf 1783  wcel 2108  wne 2940  wss 3951  c0 4333  cmpt 5225   Or wor 5591  ran crn 5686  Fincfn 8985  supcsup 9480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-en 8986  df-dom 8987  df-fin 8989  df-sup 9482
This theorem is referenced by:  uzublem  45441  limsupubuzlem  45727
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