Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > qshash | Structured version Visualization version GIF version |
Description: The cardinality of a set with an equivalence relation is the sum of the cardinalities of its equivalence classes. (Contributed by Mario Carneiro, 16-Jan-2015.) |
Ref | Expression |
---|---|
qshash.1 | ⊢ (𝜑 → ∼ Er 𝐴) |
qshash.2 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
Ref | Expression |
---|---|
qshash | ⊢ (𝜑 → (♯‘𝐴) = Σ𝑥 ∈ (𝐴 / ∼ )(♯‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qshash.1 | . . . 4 ⊢ (𝜑 → ∼ Er 𝐴) | |
2 | qshash.2 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
3 | erex 8302 | . . . . 5 ⊢ ( ∼ Er 𝐴 → (𝐴 ∈ Fin → ∼ ∈ V)) | |
4 | 1, 2, 3 | sylc 65 | . . . 4 ⊢ (𝜑 → ∼ ∈ V) |
5 | 1, 4 | uniqs2 8348 | . . 3 ⊢ (𝜑 → ∪ (𝐴 / ∼ ) = 𝐴) |
6 | 5 | fveq2d 6667 | . 2 ⊢ (𝜑 → (♯‘∪ (𝐴 / ∼ )) = (♯‘𝐴)) |
7 | pwfi 8807 | . . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin) | |
8 | 2, 7 | sylib 219 | . . . 4 ⊢ (𝜑 → 𝒫 𝐴 ∈ Fin) |
9 | 1 | qsss 8347 | . . . 4 ⊢ (𝜑 → (𝐴 / ∼ ) ⊆ 𝒫 𝐴) |
10 | 8, 9 | ssfid 8729 | . . 3 ⊢ (𝜑 → (𝐴 / ∼ ) ∈ Fin) |
11 | elpwi 4547 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴) | |
12 | ssfi 8726 | . . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ⊆ 𝐴) → 𝑥 ∈ Fin) | |
13 | 12 | ex 413 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (𝑥 ⊆ 𝐴 → 𝑥 ∈ Fin)) |
14 | 2, 11, 13 | syl2im 40 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ Fin)) |
15 | 14 | ssrdv 3970 | . . . 4 ⊢ (𝜑 → 𝒫 𝐴 ⊆ Fin) |
16 | 9, 15 | sstrd 3974 | . . 3 ⊢ (𝜑 → (𝐴 / ∼ ) ⊆ Fin) |
17 | qsdisj2 8364 | . . . 4 ⊢ ( ∼ Er 𝐴 → Disj 𝑥 ∈ (𝐴 / ∼ )𝑥) | |
18 | 1, 17 | syl 17 | . . 3 ⊢ (𝜑 → Disj 𝑥 ∈ (𝐴 / ∼ )𝑥) |
19 | 10, 16, 18 | hashuni 15169 | . 2 ⊢ (𝜑 → (♯‘∪ (𝐴 / ∼ )) = Σ𝑥 ∈ (𝐴 / ∼ )(♯‘𝑥)) |
20 | 6, 19 | eqtr3d 2855 | 1 ⊢ (𝜑 → (♯‘𝐴) = Σ𝑥 ∈ (𝐴 / ∼ )(♯‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 𝒫 cpw 4535 ∪ cuni 4830 Disj wdisj 5022 ‘cfv 6348 Er wer 8275 / cqs 8277 Fincfn 8497 ♯chash 13678 Σcsu 15030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-disj 5023 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-ec 8280 df-qs 8284 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-sum 15031 |
This theorem is referenced by: lagsubg2 18279 sylow1lem3 18654 sylow2a 18673 hashclwwlkn0 27780 |
Copyright terms: Public domain | W3C validator |