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Mirrors > Home > MPE Home > Th. List > Mathboxes > erdszelem2 | Structured version Visualization version GIF version |
Description: Lemma for erdsze 31184. (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 12771 | . . . . 5 ⊢ (1...𝐴) ∈ Fin | |
2 | pwfi 8261 | . . . . 5 ⊢ ((1...𝐴) ∈ Fin ↔ 𝒫 (1...𝐴) ∈ Fin) | |
3 | 1, 2 | mpbi 220 | . . . 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 3687 | . . . . 5 ⊢ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} ⊆ 𝒫 (1...𝐴) | |
6 | 4, 5 | eqsstri 3635 | . . . 4 ⊢ 𝑆 ⊆ 𝒫 (1...𝐴) |
7 | ssfi 8180 | . . . 4 ⊢ ((𝒫 (1...𝐴) ∈ Fin ∧ 𝑆 ⊆ 𝒫 (1...𝐴)) → 𝑆 ∈ Fin) | |
8 | 3, 6, 7 | mp2an 708 | . . 3 ⊢ 𝑆 ∈ Fin |
9 | hashf 13125 | . . . . 5 ⊢ #:V⟶(ℕ0 ∪ {+∞}) | |
10 | ffun 6048 | . . . . 5 ⊢ (#:V⟶(ℕ0 ∪ {+∞}) → Fun #) | |
11 | 9, 10 | ax-mp 5 | . . . 4 ⊢ Fun # |
12 | ssv 3625 | . . . . 5 ⊢ 𝑆 ⊆ V | |
13 | 9 | fdmi 6052 | . . . . 5 ⊢ dom # = V |
14 | 12, 13 | sseqtr4i 3638 | . . . 4 ⊢ 𝑆 ⊆ dom # |
15 | fores 6124 | . . . 4 ⊢ ((Fun # ∧ 𝑆 ⊆ dom #) → (# ↾ 𝑆):𝑆–onto→(# “ 𝑆)) | |
16 | 11, 14, 15 | mp2an 708 | . . 3 ⊢ (# ↾ 𝑆):𝑆–onto→(# “ 𝑆) |
17 | fofi 8252 | . . 3 ⊢ ((𝑆 ∈ Fin ∧ (# ↾ 𝑆):𝑆–onto→(# “ 𝑆)) → (# “ 𝑆) ∈ Fin) | |
18 | 8, 16, 17 | mp2an 708 | . 2 ⊢ (# “ 𝑆) ∈ Fin |
19 | funimass4 6247 | . . . 4 ⊢ ((Fun # ∧ 𝑆 ⊆ dom #) → ((# “ 𝑆) ⊆ ℕ ↔ ∀𝑥 ∈ 𝑆 (#‘𝑥) ∈ ℕ)) | |
20 | 11, 14, 19 | mp2an 708 | . . 3 ⊢ ((# “ 𝑆) ⊆ ℕ ↔ ∀𝑥 ∈ 𝑆 (#‘𝑥) ∈ ℕ) |
21 | 4 | erdszelem1 31173 | . . . 4 ⊢ (𝑥 ∈ 𝑆 ↔ (𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥)) |
22 | ne0i 3921 | . . . . . 6 ⊢ (𝐴 ∈ 𝑥 → 𝑥 ≠ ∅) | |
23 | 22 | 3ad2ant3 1084 | . . . . 5 ⊢ ((𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥) → 𝑥 ≠ ∅) |
24 | simp1 1061 | . . . . . . 7 ⊢ ((𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥) → 𝑥 ⊆ (1...𝐴)) | |
25 | ssfi 8180 | . . . . . . 7 ⊢ (((1...𝐴) ∈ Fin ∧ 𝑥 ⊆ (1...𝐴)) → 𝑥 ∈ Fin) | |
26 | 1, 24, 25 | sylancr 695 | . . . . . 6 ⊢ ((𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥) → 𝑥 ∈ Fin) |
27 | hashnncl 13157 | . . . . . 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 247 | . . . 4 ⊢ ((𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥) → (#‘𝑥) ∈ ℕ) |
30 | 21, 29 | sylbi 207 | . . 3 ⊢ (𝑥 ∈ 𝑆 → (#‘𝑥) ∈ ℕ) |
31 | 20, 30 | mprgbir 2927 | . 2 ⊢ (# “ 𝑆) ⊆ ℕ |
32 | 18, 31 | pm3.2i 471 | 1 ⊢ ((# “ 𝑆) ∈ Fin ∧ (# “ 𝑆) ⊆ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 {crab 2916 Vcvv 3200 ∪ cun 3572 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 {csn 4177 dom cdm 5114 ↾ cres 5116 “ cima 5117 Fun wfun 5882 ⟶wf 5884 –onto→wfo 5886 ‘cfv 5888 Isom wiso 5889 (class class class)co 6650 Fincfn 7955 1c1 9937 +∞cpnf 10071 < clt 10074 ℕcn 11020 ℕ0cn0 11292 ...cfz 12326 #chash 13117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-hash 13118 |
This theorem is referenced by: erdszelem5 31177 erdszelem6 31178 erdszelem7 31179 erdszelem8 31180 |
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