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Mirrors > Home > MPE Home > Th. List > Mathboxes > erdszelem2 | Structured version Visualization version GIF version |
Description: Lemma for erdsze 32562. (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 13335 | . . . . 5 ⊢ (1...𝐴) ∈ Fin | |
2 | pwfi 8803 | . . . . 5 ⊢ ((1...𝐴) ∈ Fin ↔ 𝒫 (1...𝐴) ∈ Fin) | |
3 | 1, 2 | mpbi 233 | . . . 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 4007 | . . . . 5 ⊢ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} ⊆ 𝒫 (1...𝐴) | |
6 | 4, 5 | eqsstri 3949 | . . . 4 ⊢ 𝑆 ⊆ 𝒫 (1...𝐴) |
7 | ssfi 8722 | . . . 4 ⊢ ((𝒫 (1...𝐴) ∈ Fin ∧ 𝑆 ⊆ 𝒫 (1...𝐴)) → 𝑆 ∈ Fin) | |
8 | 3, 6, 7 | mp2an 691 | . . 3 ⊢ 𝑆 ∈ Fin |
9 | hashf 13694 | . . . . 5 ⊢ ♯:V⟶(ℕ0 ∪ {+∞}) | |
10 | ffun 6490 | . . . . 5 ⊢ (♯:V⟶(ℕ0 ∪ {+∞}) → Fun ♯) | |
11 | 9, 10 | ax-mp 5 | . . . 4 ⊢ Fun ♯ |
12 | ssv 3939 | . . . . 5 ⊢ 𝑆 ⊆ V | |
13 | 9 | fdmi 6498 | . . . . 5 ⊢ dom ♯ = V |
14 | 12, 13 | sseqtrri 3952 | . . . 4 ⊢ 𝑆 ⊆ dom ♯ |
15 | fores 6575 | . . . 4 ⊢ ((Fun ♯ ∧ 𝑆 ⊆ dom ♯) → (♯ ↾ 𝑆):𝑆–onto→(♯ “ 𝑆)) | |
16 | 11, 14, 15 | mp2an 691 | . . 3 ⊢ (♯ ↾ 𝑆):𝑆–onto→(♯ “ 𝑆) |
17 | fofi 8794 | . . 3 ⊢ ((𝑆 ∈ Fin ∧ (♯ ↾ 𝑆):𝑆–onto→(♯ “ 𝑆)) → (♯ “ 𝑆) ∈ Fin) | |
18 | 8, 16, 17 | mp2an 691 | . 2 ⊢ (♯ “ 𝑆) ∈ Fin |
19 | funimass4 6705 | . . . 4 ⊢ ((Fun ♯ ∧ 𝑆 ⊆ dom ♯) → ((♯ “ 𝑆) ⊆ ℕ ↔ ∀𝑥 ∈ 𝑆 (♯‘𝑥) ∈ ℕ)) | |
20 | 11, 14, 19 | mp2an 691 | . . 3 ⊢ ((♯ “ 𝑆) ⊆ ℕ ↔ ∀𝑥 ∈ 𝑆 (♯‘𝑥) ∈ ℕ) |
21 | 4 | erdszelem1 32551 | . . . 4 ⊢ (𝑥 ∈ 𝑆 ↔ (𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥)) |
22 | ne0i 4250 | . . . . . 6 ⊢ (𝐴 ∈ 𝑥 → 𝑥 ≠ ∅) | |
23 | 22 | 3ad2ant3 1132 | . . . . 5 ⊢ ((𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥) → 𝑥 ≠ ∅) |
24 | simp1 1133 | . . . . . . 7 ⊢ ((𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥) → 𝑥 ⊆ (1...𝐴)) | |
25 | ssfi 8722 | . . . . . . 7 ⊢ (((1...𝐴) ∈ Fin ∧ 𝑥 ⊆ (1...𝐴)) → 𝑥 ∈ Fin) | |
26 | 1, 24, 25 | sylancr 590 | . . . . . 6 ⊢ ((𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥) → 𝑥 ∈ Fin) |
27 | hashnncl 13723 | . . . . . 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 260 | . . . 4 ⊢ ((𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥) → (♯‘𝑥) ∈ ℕ) |
30 | 21, 29 | sylbi 220 | . . 3 ⊢ (𝑥 ∈ 𝑆 → (♯‘𝑥) ∈ ℕ) |
31 | 20, 30 | mprgbir 3121 | . 2 ⊢ (♯ “ 𝑆) ⊆ ℕ |
32 | 18, 31 | pm3.2i 474 | 1 ⊢ ((♯ “ 𝑆) ∈ Fin ∧ (♯ “ 𝑆) ⊆ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 {crab 3110 Vcvv 3441 ∪ cun 3879 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 {csn 4525 dom cdm 5519 ↾ cres 5521 “ cima 5522 Fun wfun 6318 ⟶wf 6320 –onto→wfo 6322 ‘cfv 6324 Isom wiso 6325 (class class class)co 7135 Fincfn 8492 1c1 10527 +∞cpnf 10661 < clt 10664 ℕcn 11625 ℕ0cn0 11885 ...cfz 12885 ♯chash 13686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12886 df-hash 13687 |
This theorem is referenced by: erdszelem5 32555 erdszelem6 32556 erdszelem7 32557 erdszelem8 32558 |
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