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Mirrors > Home > MPE Home > Th. List > psrbagconcl | Structured version Visualization version GIF version |
Description: The complement of a bag is a bag. (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
psrbag.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
psrbagconf1o.1 | ⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝐹} |
Ref | Expression |
---|---|
psrbagconcl | ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆) → (𝐹 ∘𝑓 − 𝑋) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1054 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆) → 𝐼 ∈ 𝑉) | |
2 | simp2 1055 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆) → 𝐹 ∈ 𝐷) | |
3 | simp3 1056 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆) | |
4 | breq1 4586 | . . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝑦 ∘𝑟 ≤ 𝐹 ↔ 𝑋 ∘𝑟 ≤ 𝐹)) | |
5 | psrbagconf1o.1 | . . . . . . 7 ⊢ 𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝐹} | |
6 | 4, 5 | elrab2 3333 | . . . . . 6 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ 𝐷 ∧ 𝑋 ∘𝑟 ≤ 𝐹)) |
7 | 3, 6 | sylib 207 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆) → (𝑋 ∈ 𝐷 ∧ 𝑋 ∘𝑟 ≤ 𝐹)) |
8 | 7 | simpld 474 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐷) |
9 | psrbag.d | . . . . 5 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
10 | 9 | psrbagf 19186 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑋 ∈ 𝐷) → 𝑋:𝐼⟶ℕ0) |
11 | 1, 8, 10 | syl2anc 691 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆) → 𝑋:𝐼⟶ℕ0) |
12 | 7 | simprd 478 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∘𝑟 ≤ 𝐹) |
13 | 9 | psrbagcon 19192 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝑋:𝐼⟶ℕ0 ∧ 𝑋 ∘𝑟 ≤ 𝐹)) → ((𝐹 ∘𝑓 − 𝑋) ∈ 𝐷 ∧ (𝐹 ∘𝑓 − 𝑋) ∘𝑟 ≤ 𝐹)) |
14 | 1, 2, 11, 12, 13 | syl13anc 1320 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘𝑓 − 𝑋) ∈ 𝐷 ∧ (𝐹 ∘𝑓 − 𝑋) ∘𝑟 ≤ 𝐹)) |
15 | breq1 4586 | . . 3 ⊢ (𝑦 = (𝐹 ∘𝑓 − 𝑋) → (𝑦 ∘𝑟 ≤ 𝐹 ↔ (𝐹 ∘𝑓 − 𝑋) ∘𝑟 ≤ 𝐹)) | |
16 | 15, 5 | elrab2 3333 | . 2 ⊢ ((𝐹 ∘𝑓 − 𝑋) ∈ 𝑆 ↔ ((𝐹 ∘𝑓 − 𝑋) ∈ 𝐷 ∧ (𝐹 ∘𝑓 − 𝑋) ∘𝑟 ≤ 𝐹)) |
17 | 14, 16 | sylibr 223 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆) → (𝐹 ∘𝑓 − 𝑋) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {crab 2900 class class class wbr 4583 ◡ccnv 5037 “ cima 5041 ⟶wf 5800 (class class class)co 6549 ∘𝑓 cof 6793 ∘𝑟 cofr 6794 ↑𝑚 cmap 7744 Fincfn 7841 ≤ cle 9954 − cmin 10145 ℕcn 10897 ℕ0cn0 11169 |
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-rep 4699 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-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-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-ofr 6796 df-om 6958 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 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 |
This theorem is referenced by: psrass1lem 19198 psrdi 19227 psrdir 19228 psrass23l 19229 psrcom 19230 psrass23 19231 resspsrmul 19238 mplsubrglem 19260 mplmonmul 19285 psropprmul 19429 mdegmullem 23642 |
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