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Mirrors > Home > MPE Home > Th. List > Mathboxes > erdszelem2 | Structured version Visualization version GIF version |
Description: Lemma for erdsze 30438. (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 12633 | . . . . 5 ⊢ (1...𝐴) ∈ Fin | |
2 | pwfi 8144 | . . . . 5 ⊢ ((1...𝐴) ∈ Fin ↔ 𝒫 (1...𝐴) ∈ Fin) | |
3 | 1, 2 | mpbi 219 | . . . 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 3650 | . . . . 5 ⊢ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} ⊆ 𝒫 (1...𝐴) | |
6 | 4, 5 | eqsstri 3598 | . . . 4 ⊢ 𝑆 ⊆ 𝒫 (1...𝐴) |
7 | ssfi 8065 | . . . 4 ⊢ ((𝒫 (1...𝐴) ∈ Fin ∧ 𝑆 ⊆ 𝒫 (1...𝐴)) → 𝑆 ∈ Fin) | |
8 | 3, 6, 7 | mp2an 704 | . . 3 ⊢ 𝑆 ∈ Fin |
9 | hashf 12987 | . . . . 5 ⊢ #:V⟶(ℕ0 ∪ {+∞}) | |
10 | ffun 5961 | . . . . 5 ⊢ (#:V⟶(ℕ0 ∪ {+∞}) → Fun #) | |
11 | 9, 10 | ax-mp 5 | . . . 4 ⊢ Fun # |
12 | ssv 3588 | . . . . 5 ⊢ 𝑆 ⊆ V | |
13 | 9 | fdmi 5965 | . . . . 5 ⊢ dom # = V |
14 | 12, 13 | sseqtr4i 3601 | . . . 4 ⊢ 𝑆 ⊆ dom # |
15 | fores 6037 | . . . 4 ⊢ ((Fun # ∧ 𝑆 ⊆ dom #) → (# ↾ 𝑆):𝑆–onto→(# “ 𝑆)) | |
16 | 11, 14, 15 | mp2an 704 | . . 3 ⊢ (# ↾ 𝑆):𝑆–onto→(# “ 𝑆) |
17 | fofi 8135 | . . 3 ⊢ ((𝑆 ∈ Fin ∧ (# ↾ 𝑆):𝑆–onto→(# “ 𝑆)) → (# “ 𝑆) ∈ Fin) | |
18 | 8, 16, 17 | mp2an 704 | . 2 ⊢ (# “ 𝑆) ∈ Fin |
19 | funimass4 6157 | . . . 4 ⊢ ((Fun # ∧ 𝑆 ⊆ dom #) → ((# “ 𝑆) ⊆ ℕ ↔ ∀𝑥 ∈ 𝑆 (#‘𝑥) ∈ ℕ)) | |
20 | 11, 14, 19 | mp2an 704 | . . 3 ⊢ ((# “ 𝑆) ⊆ ℕ ↔ ∀𝑥 ∈ 𝑆 (#‘𝑥) ∈ ℕ) |
21 | 4 | erdszelem1 30427 | . . . 4 ⊢ (𝑥 ∈ 𝑆 ↔ (𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥)) |
22 | ne0i 3880 | . . . . . 6 ⊢ (𝐴 ∈ 𝑥 → 𝑥 ≠ ∅) | |
23 | 22 | 3ad2ant3 1077 | . . . . 5 ⊢ ((𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥) → 𝑥 ≠ ∅) |
24 | simp1 1054 | . . . . . . 7 ⊢ ((𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥) → 𝑥 ⊆ (1...𝐴)) | |
25 | ssfi 8065 | . . . . . . 7 ⊢ (((1...𝐴) ∈ Fin ∧ 𝑥 ⊆ (1...𝐴)) → 𝑥 ∈ Fin) | |
26 | 1, 24, 25 | sylancr 694 | . . . . . 6 ⊢ ((𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥) → 𝑥 ∈ Fin) |
27 | hashnncl 13018 | . . . . . 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 246 | . . . 4 ⊢ ((𝑥 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑥) Isom < , 𝑂 (𝑥, (𝐹 “ 𝑥)) ∧ 𝐴 ∈ 𝑥) → (#‘𝑥) ∈ ℕ) |
30 | 21, 29 | sylbi 206 | . . 3 ⊢ (𝑥 ∈ 𝑆 → (#‘𝑥) ∈ ℕ) |
31 | 20, 30 | mprgbir 2911 | . 2 ⊢ (# “ 𝑆) ⊆ ℕ |
32 | 18, 31 | pm3.2i 470 | 1 ⊢ ((# “ 𝑆) ∈ Fin ∧ (# “ 𝑆) ⊆ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 {crab 2900 Vcvv 3173 ∪ cun 3538 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 dom cdm 5038 ↾ cres 5040 “ cima 5041 Fun wfun 5798 ⟶wf 5800 –onto→wfo 5802 ‘cfv 5804 Isom wiso 5805 (class class class)co 6549 Fincfn 7841 1c1 9816 +∞cpnf 9950 < clt 9953 ℕcn 10897 ℕ0cn0 11169 ...cfz 12197 #chash 12979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 |
This theorem is referenced by: erdszelem5 30431 erdszelem6 30432 erdszelem7 30433 erdszelem8 30434 |
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