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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 2823 | . . 3 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
3 | fisupclrnmpt.c | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝐴) | |
4 | 1, 2, 3 | rnmptssd 41465 | . 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 41434 | . . . 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 41491 | . . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≠ ∅) |
11 | fisupcl 8935 | . . 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 1368 | . 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐵 ↦ 𝐶), 𝐴, 𝑅) ∈ ran (𝑥 ∈ 𝐵 ↦ 𝐶)) |
13 | 4, 12 | sseldd 3970 | 1 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐵 ↦ 𝐶), 𝐴, 𝑅) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 Ⅎwnf 1784 ∈ wcel 2114 ≠ wne 3018 ⊆ wss 3938 ∅c0 4293 ↦ cmpt 5148 Or wor 5475 ran crn 5558 Fincfn 8511 supcsup 8906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-fin 8515 df-sup 8908 |
This theorem is referenced by: uzublem 41711 limsupubuzlem 42000 |
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