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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 2825 | . . 3 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
3 | fisupclrnmpt.c | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝐴) | |
4 | 1, 2, 3 | rnmptssd 40193 | . 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 40160 | . . . 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 40219 | . . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≠ ∅) |
11 | fisupcl 8644 | . . 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 1497 | . 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐵 ↦ 𝐶), 𝐴, 𝑅) ∈ ran (𝑥 ∈ 𝐵 ↦ 𝐶)) |
13 | 4, 12 | sseldd 3828 | 1 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐵 ↦ 𝐶), 𝐴, 𝑅) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 Ⅎwnf 1884 ∈ wcel 2166 ≠ wne 2999 ⊆ wss 3798 ∅c0 4144 ↦ cmpt 4952 Or wor 5262 ran crn 5343 Fincfn 8222 supcsup 8615 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-en 8223 df-dom 8224 df-fin 8226 df-sup 8617 |
This theorem is referenced by: uzublem 40452 limsupubuzlem 40739 |
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