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Mirrors > Home > MPE Home > Th. List > Mathboxes > erdszelem2 | Structured version Visualization version GIF version |
Description: Lemma for erdsze 32346. (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 13328 | . . . . 5 ⊢ (1...𝐴) ∈ Fin | |
2 | pwfi 8807 | . . . . 5 ⊢ ((1...𝐴) ∈ Fin ↔ 𝒫 (1...𝐴) ∈ Fin) | |
3 | 1, 2 | mpbi 231 | . . . 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 4053 | . . . . 5 ⊢ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} ⊆ 𝒫 (1...𝐴) | |
6 | 4, 5 | eqsstri 3998 | . . . 4 ⊢ 𝑆 ⊆ 𝒫 (1...𝐴) |
7 | ssfi 8726 | . . . 4 ⊢ ((𝒫 (1...𝐴) ∈ Fin ∧ 𝑆 ⊆ 𝒫 (1...𝐴)) → 𝑆 ∈ Fin) | |
8 | 3, 6, 7 | mp2an 688 | . . 3 ⊢ 𝑆 ∈ Fin |
9 | hashf 13686 | . . . . 5 ⊢ ♯:V⟶(ℕ0 ∪ {+∞}) | |
10 | ffun 6510 | . . . . 5 ⊢ (♯:V⟶(ℕ0 ∪ {+∞}) → Fun ♯) | |
11 | 9, 10 | ax-mp 5 | . . . 4 ⊢ Fun ♯ |
12 | ssv 3988 | . . . . 5 ⊢ 𝑆 ⊆ V | |
13 | 9 | fdmi 6517 | . . . . 5 ⊢ dom ♯ = V |
14 | 12, 13 | sseqtrri 4001 | . . . 4 ⊢ 𝑆 ⊆ dom ♯ |
15 | fores 6593 | . . . 4 ⊢ ((Fun ♯ ∧ 𝑆 ⊆ dom ♯) → (♯ ↾ 𝑆):𝑆–onto→(♯ “ 𝑆)) | |
16 | 11, 14, 15 | mp2an 688 | . . 3 ⊢ (♯ ↾ 𝑆):𝑆–onto→(♯ “ 𝑆) |
17 | fofi 8798 | . . 3 ⊢ ((𝑆 ∈ Fin ∧ (♯ ↾ 𝑆):𝑆–onto→(♯ “ 𝑆)) → (♯ “ 𝑆) ∈ Fin) | |
18 | 8, 16, 17 | mp2an 688 | . 2 ⊢ (♯ “ 𝑆) ∈ Fin |
19 | funimass4 6723 | . . . 4 ⊢ ((Fun ♯ ∧ 𝑆 ⊆ dom ♯) → ((♯ “ 𝑆) ⊆ ℕ ↔ ∀𝑥 ∈ 𝑆 (♯‘𝑥) ∈ ℕ)) | |
20 | 11, 14, 19 | mp2an 688 | . . 3 ⊢ ((♯ “ 𝑆) ⊆ ℕ ↔ ∀𝑥 ∈ 𝑆 (♯‘𝑥) ∈ ℕ) |
21 | 4 | erdszelem1 32335 | . . . 4 ⊢ (𝑥 ∈ 𝑆 ↔ (𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥)) |
22 | ne0i 4297 | . . . . . 6 ⊢ (𝐴 ∈ 𝑥 → 𝑥 ≠ ∅) | |
23 | 22 | 3ad2ant3 1127 | . . . . 5 ⊢ ((𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥) → 𝑥 ≠ ∅) |
24 | simp1 1128 | . . . . . . 7 ⊢ ((𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥) → 𝑥 ⊆ (1...𝐴)) | |
25 | ssfi 8726 | . . . . . . 7 ⊢ (((1...𝐴) ∈ Fin ∧ 𝑥 ⊆ (1...𝐴)) → 𝑥 ∈ Fin) | |
26 | 1, 24, 25 | sylancr 587 | . . . . . 6 ⊢ ((𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥) → 𝑥 ∈ Fin) |
27 | hashnncl 13715 | . . . . . 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 258 | . . . 4 ⊢ ((𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥) → (♯‘𝑥) ∈ ℕ) |
30 | 21, 29 | sylbi 218 | . . 3 ⊢ (𝑥 ∈ 𝑆 → (♯‘𝑥) ∈ ℕ) |
31 | 20, 30 | mprgbir 3150 | . 2 ⊢ (♯ “ 𝑆) ⊆ ℕ |
32 | 18, 31 | pm3.2i 471 | 1 ⊢ ((♯ “ 𝑆) ∈ Fin ∧ (♯ “ 𝑆) ⊆ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 {crab 3139 Vcvv 3492 ∪ cun 3931 ⊆ wss 3933 ∅c0 4288 𝒫 cpw 4535 {csn 4557 dom cdm 5548 ↾ cres 5550 “ cima 5551 Fun wfun 6342 ⟶wf 6344 –onto→wfo 6346 ‘cfv 6348 Isom wiso 6349 (class class class)co 7145 Fincfn 8497 1c1 10526 +∞cpnf 10660 < clt 10663 ℕcn 11626 ℕ0cn0 11885 ...cfz 12880 ♯chash 13678 |
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-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 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 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 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-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-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-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-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 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-nn 11627 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12881 df-hash 13679 |
This theorem is referenced by: erdszelem5 32339 erdszelem6 32340 erdszelem7 32341 erdszelem8 32342 |
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