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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hashunif | Structured version Visualization version GIF version |
Description: The cardinality of a disjoint finite union of finite sets. Cf. hashuni 14777. (Contributed by Thierry Arnoux, 17-Feb-2017.) |
Ref | Expression |
---|---|
hashiunf.1 | ⊢ Ⅎ𝑥𝜑 |
hashiunf.3 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
hashunif.4 | ⊢ (𝜑 → 𝐴 ⊆ Fin) |
hashunif.5 | ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝑥) |
Ref | Expression |
---|---|
hashunif | ⊢ (𝜑 → (♯‘∪ 𝐴) = Σ𝑥 ∈ 𝐴 (♯‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniiun 4725 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 | |
2 | 1 | fveq2i 6356 | . 2 ⊢ (♯‘∪ 𝐴) = (♯‘∪ 𝑥 ∈ 𝐴 𝑥) |
3 | hashiunf.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
4 | hashunif.4 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ Fin) | |
5 | 4 | sselda 3744 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ Fin) |
6 | hashunif.5 | . . . . 5 ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝑥) | |
7 | id 22 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
8 | 7 | cbvdisjv 4783 | . . . . 5 ⊢ (Disj 𝑥 ∈ 𝐴 𝑥 ↔ Disj 𝑦 ∈ 𝐴 𝑦) |
9 | 6, 8 | sylib 208 | . . . 4 ⊢ (𝜑 → Disj 𝑦 ∈ 𝐴 𝑦) |
10 | 3, 5, 9 | hashiun 14773 | . . 3 ⊢ (𝜑 → (♯‘∪ 𝑦 ∈ 𝐴 𝑦) = Σ𝑦 ∈ 𝐴 (♯‘𝑦)) |
11 | 7 | cbviunv 4711 | . . . . 5 ⊢ ∪ 𝑥 ∈ 𝐴 𝑥 = ∪ 𝑦 ∈ 𝐴 𝑦 |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝑥 = ∪ 𝑦 ∈ 𝐴 𝑦) |
13 | 12 | fveq2d 6357 | . . 3 ⊢ (𝜑 → (♯‘∪ 𝑥 ∈ 𝐴 𝑥) = (♯‘∪ 𝑦 ∈ 𝐴 𝑦)) |
14 | fveq2 6353 | . . . . 5 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦)) | |
15 | 14 | cbvsumv 14645 | . . . 4 ⊢ Σ𝑥 ∈ 𝐴 (♯‘𝑥) = Σ𝑦 ∈ 𝐴 (♯‘𝑦) |
16 | 15 | a1i 11 | . . 3 ⊢ (𝜑 → Σ𝑥 ∈ 𝐴 (♯‘𝑥) = Σ𝑦 ∈ 𝐴 (♯‘𝑦)) |
17 | 10, 13, 16 | 3eqtr4d 2804 | . 2 ⊢ (𝜑 → (♯‘∪ 𝑥 ∈ 𝐴 𝑥) = Σ𝑥 ∈ 𝐴 (♯‘𝑥)) |
18 | 2, 17 | syl5eq 2806 | 1 ⊢ (𝜑 → (♯‘∪ 𝐴) = Σ𝑥 ∈ 𝐴 (♯‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 Ⅎwnf 1857 ∈ wcel 2139 ⊆ wss 3715 ∪ cuni 4588 ∪ ciun 4672 Disj wdisj 4772 ‘cfv 6049 Fincfn 8123 ♯chash 13331 Σcsu 14635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-inf2 8713 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 ax-pre-sup 10226 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-disj 4773 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-1st 7334 df-2nd 7335 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-oadd 7734 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-sup 8515 df-oi 8582 df-card 8975 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-div 10897 df-nn 11233 df-2 11291 df-3 11292 df-n0 11505 df-z 11590 df-uz 11900 df-rp 12046 df-fz 12540 df-fzo 12680 df-seq 13016 df-exp 13075 df-hash 13332 df-cj 14058 df-re 14059 df-im 14060 df-sqrt 14194 df-abs 14195 df-clim 14438 df-sum 14636 |
This theorem is referenced by: hasheuni 30477 |
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