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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 2733 | . . 3 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
3 | fisupclrnmpt.c | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝐴) | |
4 | 1, 2, 3 | rnmptssd 43504 | . 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 43476 | . . . 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 6197 | . . 3 ⊢ (𝜑 → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≠ ∅) |
11 | fisupcl 9410 | . . 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 1373 | . 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐵 ↦ 𝐶), 𝐴, 𝑅) ∈ ran (𝑥 ∈ 𝐵 ↦ 𝐶)) |
13 | 4, 12 | sseldd 3946 | 1 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐵 ↦ 𝐶), 𝐴, 𝑅) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 Ⅎwnf 1786 ∈ wcel 2107 ≠ wne 2940 ⊆ wss 3911 ∅c0 4283 ↦ cmpt 5189 Or wor 5545 ran crn 5635 Fincfn 8886 supcsup 9381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-om 7804 df-1st 7922 df-2nd 7923 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-fin 8890 df-sup 9383 |
This theorem is referenced by: uzublem 43751 limsupubuzlem 44039 |
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