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Theorem fisupclrnmpt 44839
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 2725 . . 3 (𝑥𝐵𝐶) = (𝑥𝐵𝐶)
3 fisupclrnmpt.c . . 3 ((𝜑𝑥𝐵) → 𝐶𝐴)
41, 2, 3rnmptssd 44629 . 2 (𝜑 → ran (𝑥𝐵𝐶) ⊆ 𝐴)
5 fisupclrnmpt.r . . 3 (𝜑𝑅 Or 𝐴)
6 fisupclrnmpt.b . . . 4 (𝜑𝐵 ∈ Fin)
72rnmptfi 44604 . . . 4 (𝐵 ∈ Fin → ran (𝑥𝐵𝐶) ∈ Fin)
86, 7syl 17 . . 3 (𝜑 → ran (𝑥𝐵𝐶) ∈ Fin)
9 fisupclrnmpt.n . . . 4 (𝜑𝐵 ≠ ∅)
101, 3, 2, 9rnmptn0 6244 . . 3 (𝜑 → ran (𝑥𝐵𝐶) ≠ ∅)
11 fisupcl 9487 . . 3 ((𝑅 Or 𝐴 ∧ (ran (𝑥𝐵𝐶) ∈ Fin ∧ ran (𝑥𝐵𝐶) ≠ ∅ ∧ ran (𝑥𝐵𝐶) ⊆ 𝐴)) → sup(ran (𝑥𝐵𝐶), 𝐴, 𝑅) ∈ ran (𝑥𝐵𝐶))
125, 8, 10, 4, 11syl13anc 1369 . 2 (𝜑 → sup(ran (𝑥𝐵𝐶), 𝐴, 𝑅) ∈ ran (𝑥𝐵𝐶))
134, 12sseldd 3974 1 (𝜑 → sup(ran (𝑥𝐵𝐶), 𝐴, 𝑅) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wnf 1777  wcel 2098  wne 2930  wss 3941  c0 4319  cmpt 5227   Or wor 5584  ran crn 5674  Fincfn 8957  supcsup 9458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-om 7866  df-1st 7987  df-2nd 7988  df-1o 8480  df-er 8718  df-en 8958  df-dom 8959  df-fin 8961  df-sup 9460
This theorem is referenced by:  uzublem  44871  limsupubuzlem  45159
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