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Mirrors > Home > MPE Home > Th. List > hashrabrex | Structured version Visualization version GIF version |
Description: The number of elements in a class abstraction with a restricted existential quantification. (Contributed by Alexander van der Vekens, 29-Jul-2018.) |
Ref | Expression |
---|---|
hashrabrex.1 | ⊢ (𝜑 → 𝑌 ∈ Fin) |
hashrabrex.2 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → {𝑥 ∈ 𝑋 ∣ 𝜓} ∈ Fin) |
hashrabrex.3 | ⊢ (𝜑 → Disj 𝑦 ∈ 𝑌 {𝑥 ∈ 𝑋 ∣ 𝜓}) |
Ref | Expression |
---|---|
hashrabrex | ⊢ (𝜑 → (♯‘{𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑌 𝜓}) = Σ𝑦 ∈ 𝑌 (♯‘{𝑥 ∈ 𝑋 ∣ 𝜓})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunrab 4756 | . . . 4 ⊢ ∪ 𝑦 ∈ 𝑌 {𝑥 ∈ 𝑋 ∣ 𝜓} = {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑌 𝜓} | |
2 | 1 | eqcomi 2807 | . . 3 ⊢ {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑌 𝜓} = ∪ 𝑦 ∈ 𝑌 {𝑥 ∈ 𝑋 ∣ 𝜓} |
3 | 2 | fveq2i 6413 | . 2 ⊢ (♯‘{𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑌 𝜓}) = (♯‘∪ 𝑦 ∈ 𝑌 {𝑥 ∈ 𝑋 ∣ 𝜓}) |
4 | hashrabrex.1 | . . 3 ⊢ (𝜑 → 𝑌 ∈ Fin) | |
5 | hashrabrex.2 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → {𝑥 ∈ 𝑋 ∣ 𝜓} ∈ Fin) | |
6 | hashrabrex.3 | . . 3 ⊢ (𝜑 → Disj 𝑦 ∈ 𝑌 {𝑥 ∈ 𝑋 ∣ 𝜓}) | |
7 | 4, 5, 6 | hashiun 14889 | . 2 ⊢ (𝜑 → (♯‘∪ 𝑦 ∈ 𝑌 {𝑥 ∈ 𝑋 ∣ 𝜓}) = Σ𝑦 ∈ 𝑌 (♯‘{𝑥 ∈ 𝑋 ∣ 𝜓})) |
8 | 3, 7 | syl5eq 2844 | 1 ⊢ (𝜑 → (♯‘{𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑌 𝜓}) = Σ𝑦 ∈ 𝑌 (♯‘{𝑥 ∈ 𝑋 ∣ 𝜓})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∃wrex 3089 {crab 3092 ∪ ciun 4709 Disj wdisj 4810 ‘cfv 6100 Fincfn 8194 ♯chash 13367 Σcsu 14754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-rep 4963 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 ax-inf2 8787 ax-cnex 10279 ax-resscn 10280 ax-1cn 10281 ax-icn 10282 ax-addcl 10283 ax-addrcl 10284 ax-mulcl 10285 ax-mulrcl 10286 ax-mulcom 10287 ax-addass 10288 ax-mulass 10289 ax-distr 10290 ax-i2m1 10291 ax-1ne0 10292 ax-1rid 10293 ax-rnegex 10294 ax-rrecex 10295 ax-cnre 10296 ax-pre-lttri 10297 ax-pre-lttrn 10298 ax-pre-ltadd 10299 ax-pre-mulgt0 10300 ax-pre-sup 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-pss 3784 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-tp 4372 df-op 4374 df-uni 4628 df-int 4667 df-iun 4711 df-disj 4811 df-br 4843 df-opab 4905 df-mpt 4922 df-tr 4945 df-id 5219 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-se 5271 df-we 5272 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-pred 5897 df-ord 5943 df-on 5944 df-lim 5945 df-suc 5946 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-isom 6109 df-riota 6838 df-ov 6880 df-oprab 6881 df-mpt2 6882 df-om 7299 df-1st 7400 df-2nd 7401 df-wrecs 7644 df-recs 7706 df-rdg 7744 df-1o 7798 df-oadd 7802 df-er 7981 df-en 8195 df-dom 8196 df-sdom 8197 df-fin 8198 df-sup 8589 df-oi 8656 df-card 9050 df-pnf 10364 df-mnf 10365 df-xr 10366 df-ltxr 10367 df-le 10368 df-sub 10557 df-neg 10558 df-div 10976 df-nn 11312 df-2 11373 df-3 11374 df-n0 11578 df-z 11664 df-uz 11928 df-rp 12072 df-fz 12578 df-fzo 12718 df-seq 13053 df-exp 13112 df-hash 13368 df-cj 14177 df-re 14178 df-im 14179 df-sqrt 14313 df-abs 14314 df-clim 14557 df-sum 14755 |
This theorem is referenced by: hashwwlksnext 27193 hashwwlksnextOLD 27194 |
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