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Mirrors > Home > MPE Home > Th. List > Mathboxes > erdszelem2 | Structured version Visualization version GIF version |
Description: Lemma for erdsze 31730. (Contributed by Mario Carneiro, 22-Jan-2015.) |
Ref | Expression |
---|---|
erdszelem1.1 | ⊢ 𝑆 = {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} |
Ref | Expression |
---|---|
erdszelem2 | ⊢ ((♯ “ 𝑆) ∈ Fin ∧ (♯ “ 𝑆) ⊆ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzfi 13066 | . . . . 5 ⊢ (1...𝐴) ∈ Fin | |
2 | pwfi 8530 | . . . . 5 ⊢ ((1...𝐴) ∈ Fin ↔ 𝒫 (1...𝐴) ∈ Fin) | |
3 | 1, 2 | mpbi 222 | . . . 4 ⊢ 𝒫 (1...𝐴) ∈ Fin |
4 | erdszelem1.1 | . . . . 5 ⊢ 𝑆 = {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} | |
5 | ssrab2 3912 | . . . . 5 ⊢ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} ⊆ 𝒫 (1...𝐴) | |
6 | 4, 5 | eqsstri 3860 | . . . 4 ⊢ 𝑆 ⊆ 𝒫 (1...𝐴) |
7 | ssfi 8449 | . . . 4 ⊢ ((𝒫 (1...𝐴) ∈ Fin ∧ 𝑆 ⊆ 𝒫 (1...𝐴)) → 𝑆 ∈ Fin) | |
8 | 3, 6, 7 | mp2an 685 | . . 3 ⊢ 𝑆 ∈ Fin |
9 | hashf 13418 | . . . . 5 ⊢ ♯:V⟶(ℕ0 ∪ {+∞}) | |
10 | ffun 6281 | . . . . 5 ⊢ (♯:V⟶(ℕ0 ∪ {+∞}) → Fun ♯) | |
11 | 9, 10 | ax-mp 5 | . . . 4 ⊢ Fun ♯ |
12 | ssv 3850 | . . . . 5 ⊢ 𝑆 ⊆ V | |
13 | 9 | fdmi 6288 | . . . . 5 ⊢ dom ♯ = V |
14 | 12, 13 | sseqtr4i 3863 | . . . 4 ⊢ 𝑆 ⊆ dom ♯ |
15 | fores 6363 | . . . 4 ⊢ ((Fun ♯ ∧ 𝑆 ⊆ dom ♯) → (♯ ↾ 𝑆):𝑆–onto→(♯ “ 𝑆)) | |
16 | 11, 14, 15 | mp2an 685 | . . 3 ⊢ (♯ ↾ 𝑆):𝑆–onto→(♯ “ 𝑆) |
17 | fofi 8521 | . . 3 ⊢ ((𝑆 ∈ Fin ∧ (♯ ↾ 𝑆):𝑆–onto→(♯ “ 𝑆)) → (♯ “ 𝑆) ∈ Fin) | |
18 | 8, 16, 17 | mp2an 685 | . 2 ⊢ (♯ “ 𝑆) ∈ Fin |
19 | funimass4 6494 | . . . 4 ⊢ ((Fun ♯ ∧ 𝑆 ⊆ dom ♯) → ((♯ “ 𝑆) ⊆ ℕ ↔ ∀𝑥 ∈ 𝑆 (♯‘𝑥) ∈ ℕ)) | |
20 | 11, 14, 19 | mp2an 685 | . . 3 ⊢ ((♯ “ 𝑆) ⊆ ℕ ↔ ∀𝑥 ∈ 𝑆 (♯‘𝑥) ∈ ℕ) |
21 | 4 | erdszelem1 31719 | . . . 4 ⊢ (𝑥 ∈ 𝑆 ↔ (𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥)) |
22 | ne0i 4150 | . . . . . 6 ⊢ (𝐴 ∈ 𝑥 → 𝑥 ≠ ∅) | |
23 | 22 | 3ad2ant3 1171 | . . . . 5 ⊢ ((𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥) → 𝑥 ≠ ∅) |
24 | simp1 1172 | . . . . . . 7 ⊢ ((𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥) → 𝑥 ⊆ (1...𝐴)) | |
25 | ssfi 8449 | . . . . . . 7 ⊢ (((1...𝐴) ∈ Fin ∧ 𝑥 ⊆ (1...𝐴)) → 𝑥 ∈ Fin) | |
26 | 1, 24, 25 | sylancr 583 | . . . . . 6 ⊢ ((𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥) → 𝑥 ∈ Fin) |
27 | hashnncl 13447 | . . . . . 6 ⊢ (𝑥 ∈ Fin → ((♯‘𝑥) ∈ ℕ ↔ 𝑥 ≠ ∅)) | |
28 | 26, 27 | syl 17 | . . . . 5 ⊢ ((𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥) → ((♯‘𝑥) ∈ ℕ ↔ 𝑥 ≠ ∅)) |
29 | 23, 28 | mpbird 249 | . . . 4 ⊢ ((𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥) → (♯‘𝑥) ∈ ℕ) |
30 | 21, 29 | sylbi 209 | . . 3 ⊢ (𝑥 ∈ 𝑆 → (♯‘𝑥) ∈ ℕ) |
31 | 20, 30 | mprgbir 3136 | . 2 ⊢ (♯ “ 𝑆) ⊆ ℕ |
32 | 18, 31 | pm3.2i 464 | 1 ⊢ ((♯ “ 𝑆) ∈ Fin ∧ (♯ “ 𝑆) ⊆ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ≠ wne 2999 ∀wral 3117 {crab 3121 Vcvv 3414 ∪ cun 3796 ⊆ wss 3798 ∅c0 4144 𝒫 cpw 4378 {csn 4397 dom cdm 5342 ↾ cres 5344 “ cima 5345 Fun wfun 6117 ⟶wf 6119 –onto→wfo 6121 ‘cfv 6123 Isom wiso 6124 (class class class)co 6905 Fincfn 8222 1c1 10253 +∞cpnf 10388 < clt 10391 ℕcn 11350 ℕ0cn0 11618 ...cfz 12619 ♯chash 13410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-2o 7827 df-oadd 7830 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-n0 11619 df-xnn0 11691 df-z 11705 df-uz 11969 df-fz 12620 df-hash 13411 |
This theorem is referenced by: erdszelem5 31723 erdszelem6 31724 erdszelem7 31725 erdszelem8 31726 |
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