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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 2737 | . . 3 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
3 | fisupclrnmpt.c | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝐴) | |
4 | 1, 2, 3 | rnmptssd 42408 | . 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 42380 | . . . 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 6107 | . . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≠ ∅) |
11 | fisupcl 9085 | . . 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 1374 | . 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐵 ↦ 𝐶), 𝐴, 𝑅) ∈ ran (𝑥 ∈ 𝐵 ↦ 𝐶)) |
13 | 4, 12 | sseldd 3902 | 1 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐵 ↦ 𝐶), 𝐴, 𝑅) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 Ⅎwnf 1791 ∈ wcel 2110 ≠ wne 2940 ⊆ wss 3866 ∅c0 4237 ↦ cmpt 5135 Or wor 5467 ran crn 5552 Fincfn 8626 supcsup 9056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-om 7645 df-1st 7761 df-2nd 7762 df-1o 8202 df-er 8391 df-en 8627 df-dom 8628 df-fin 8630 df-sup 9058 |
This theorem is referenced by: uzublem 42643 limsupubuzlem 42928 |
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