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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fisupclrnmpt | Structured version Visualization version GIF version |
Description: A nonempty finite indexed set contains its supremum. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
fisupclrnmpt.x | ⊢ Ⅎ𝑥𝜑 |
fisupclrnmpt.r | ⊢ (𝜑 → 𝑅 Or 𝐴) |
fisupclrnmpt.b | ⊢ (𝜑 → 𝐵 ∈ Fin) |
fisupclrnmpt.n | ⊢ (𝜑 → 𝐵 ≠ ∅) |
fisupclrnmpt.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝐴) |
Ref | Expression |
---|---|
fisupclrnmpt | ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐵 ↦ 𝐶), 𝐴, 𝑅) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fisupclrnmpt.x | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | eqid 2727 | . . 3 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
3 | fisupclrnmpt.c | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝐴) | |
4 | 1, 2, 3 | rnmptssd 44482 | . 2 ⊢ (𝜑 → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ⊆ 𝐴) |
5 | fisupclrnmpt.r | . . 3 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
6 | fisupclrnmpt.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ Fin) | |
7 | 2 | rnmptfi 44457 | . . . 4 ⊢ (𝐵 ∈ Fin → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ Fin) |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ Fin) |
9 | fisupclrnmpt.n | . . . 4 ⊢ (𝜑 → 𝐵 ≠ ∅) | |
10 | 1, 3, 2, 9 | rnmptn0 6242 | . . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≠ ∅) |
11 | fisupcl 9478 | . . 3 ⊢ ((𝑅 Or 𝐴 ∧ (ran (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ Fin ∧ ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≠ ∅ ∧ ran (𝑥 ∈ 𝐵 ↦ 𝐶) ⊆ 𝐴)) → sup(ran (𝑥 ∈ 𝐵 ↦ 𝐶), 𝐴, 𝑅) ∈ ran (𝑥 ∈ 𝐵 ↦ 𝐶)) | |
12 | 5, 8, 10, 4, 11 | syl13anc 1370 | . 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐵 ↦ 𝐶), 𝐴, 𝑅) ∈ ran (𝑥 ∈ 𝐵 ↦ 𝐶)) |
13 | 4, 12 | sseldd 3979 | 1 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐵 ↦ 𝐶), 𝐴, 𝑅) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 Ⅎwnf 1778 ∈ wcel 2099 ≠ wne 2935 ⊆ wss 3944 ∅c0 4318 ↦ cmpt 5225 Or wor 5583 ran crn 5673 Fincfn 8953 supcsup 9449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-om 7863 df-1st 7985 df-2nd 7986 df-1o 8478 df-er 8716 df-en 8954 df-dom 8955 df-fin 8957 df-sup 9451 |
This theorem is referenced by: uzublem 44725 limsupubuzlem 45013 |
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