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Mirrors > Home > MPE Home > Th. List > Mathboxes > etransclem16 | Structured version Visualization version GIF version |
Description: Every element in the range of 𝐶 is a finite set. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
etransclem16.c | ⊢ 𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛}) |
etransclem16.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
etransclem16 | ⊢ (𝜑 → (𝐶‘𝑁) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | etransclem16.c | . . 3 ⊢ 𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛}) | |
2 | etransclem16.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
3 | 1, 2 | etransclem12 43822 | . 2 ⊢ (𝜑 → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) |
4 | fzfi 13720 | . . . 4 ⊢ (0...𝑁) ∈ Fin | |
5 | fzfi 13720 | . . . 4 ⊢ (0...𝑀) ∈ Fin | |
6 | mapfi 9143 | . . . 4 ⊢ (((0...𝑁) ∈ Fin ∧ (0...𝑀) ∈ Fin) → ((0...𝑁) ↑m (0...𝑀)) ∈ Fin) | |
7 | 4, 5, 6 | mp2an 688 | . . 3 ⊢ ((0...𝑁) ↑m (0...𝑀)) ∈ Fin |
8 | ssrab2 4016 | . . 3 ⊢ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} ⊆ ((0...𝑁) ↑m (0...𝑀)) | |
9 | ssfi 8981 | . . 3 ⊢ ((((0...𝑁) ↑m (0...𝑀)) ∈ Fin ∧ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} ⊆ ((0...𝑁) ↑m (0...𝑀))) → {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} ∈ Fin) | |
10 | 7, 8, 9 | mp2an 688 | . 2 ⊢ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} ∈ Fin |
11 | 3, 10 | eqeltrdi 2842 | 1 ⊢ (𝜑 → (𝐶‘𝑁) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2101 {crab 3221 ⊆ wss 3889 ↦ cmpt 5160 ‘cfv 6447 (class class class)co 7295 ↑m cmap 8635 Fincfn 8753 0cc0 10899 ℕ0cn0 12261 ...cfz 13267 Σcsu 15425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-1st 7851 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-er 8518 df-map 8637 df-pm 8638 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-nn 12002 df-n0 12262 df-z 12348 df-uz 12611 df-fz 13268 |
This theorem is referenced by: etransclem31 43841 etransclem32 43842 etransclem34 43844 etransclem35 43845 etransclem36 43846 etransclem37 43847 etransclem38 43848 |
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