Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > caragenunicl | Structured version Visualization version GIF version |
Description: The Caratheodory's construction is closed under countable union. Step (d) in the proof of Theorem 113C of [Fremlin1] p. 20. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
caragenunicl.o | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
caragenunicl.s | ⊢ 𝑆 = (CaraGen‘𝑂) |
caragenunicl.y | ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
caragenunicl.ctb | ⊢ (𝜑 → 𝑋 ≼ ω) |
Ref | Expression |
---|---|
caragenunicl | ⊢ (𝜑 → ∪ 𝑋 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unieq 4444 | . . . . 5 ⊢ (𝑋 = ∅ → ∪ 𝑋 = ∪ ∅) | |
2 | uni0 4465 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
3 | 1, 2 | syl6eq 2672 | . . . 4 ⊢ (𝑋 = ∅ → ∪ 𝑋 = ∅) |
4 | 3 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑋 = ∅) |
5 | caragenunicl.o | . . . . 5 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
6 | caragenunicl.s | . . . . 5 ⊢ 𝑆 = (CaraGen‘𝑂) | |
7 | 5, 6 | caragen0 40720 | . . . 4 ⊢ (𝜑 → ∅ ∈ 𝑆) |
8 | 7 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ 𝑆) |
9 | 4, 8 | eqeltrd 2701 | . 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑋 ∈ 𝑆) |
10 | simpl 473 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝜑) | |
11 | neqne 2802 | . . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) | |
12 | 11 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅) |
13 | simpr 477 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ≠ ∅) | |
14 | caragenunicl.ctb | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≼ ω) | |
15 | reldom 7961 | . . . . . . . . . 10 ⊢ Rel ≼ | |
16 | brrelex 5156 | . . . . . . . . . 10 ⊢ ((Rel ≼ ∧ 𝑋 ≼ ω) → 𝑋 ∈ V) | |
17 | 15, 16 | mpan 706 | . . . . . . . . 9 ⊢ (𝑋 ≼ ω → 𝑋 ∈ V) |
18 | 14, 17 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ V) |
19 | 18 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ∈ V) |
20 | 0sdomg 8089 | . . . . . . 7 ⊢ (𝑋 ∈ V → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) | |
21 | 19, 20 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) |
22 | 13, 21 | mpbird 247 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∅ ≺ 𝑋) |
23 | nnenom 12779 | . . . . . . . . 9 ⊢ ℕ ≈ ω | |
24 | 23 | ensymi 8006 | . . . . . . . 8 ⊢ ω ≈ ℕ |
25 | 24 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ω ≈ ℕ) |
26 | domentr 8015 | . . . . . . 7 ⊢ ((𝑋 ≼ ω ∧ ω ≈ ℕ) → 𝑋 ≼ ℕ) | |
27 | 14, 25, 26 | syl2anc 693 | . . . . . 6 ⊢ (𝜑 → 𝑋 ≼ ℕ) |
28 | 27 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ≼ ℕ) |
29 | fodomr 8111 | . . . . 5 ⊢ ((∅ ≺ 𝑋 ∧ 𝑋 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto→𝑋) | |
30 | 22, 28, 29 | syl2anc 693 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∃𝑓 𝑓:ℕ–onto→𝑋) |
31 | founiiun 39360 | . . . . . . . . 9 ⊢ (𝑓:ℕ–onto→𝑋 → ∪ 𝑋 = ∪ 𝑛 ∈ ℕ (𝑓‘𝑛)) | |
32 | 31 | adantl 482 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → ∪ 𝑋 = ∪ 𝑛 ∈ ℕ (𝑓‘𝑛)) |
33 | 5 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → 𝑂 ∈ OutMeas) |
34 | 1zzd 11408 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → 1 ∈ ℤ) | |
35 | nnuz 11723 | . . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1) | |
36 | fof 6115 | . . . . . . . . . . 11 ⊢ (𝑓:ℕ–onto→𝑋 → 𝑓:ℕ⟶𝑋) | |
37 | 36 | adantl 482 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → 𝑓:ℕ⟶𝑋) |
38 | caragenunicl.y | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) | |
39 | 38 | adantr 481 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → 𝑋 ⊆ 𝑆) |
40 | 37, 39 | fssd 6057 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → 𝑓:ℕ⟶𝑆) |
41 | 33, 6, 34, 35, 40 | carageniuncl 40737 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → ∪ 𝑛 ∈ ℕ (𝑓‘𝑛) ∈ 𝑆) |
42 | 32, 41 | eqeltrd 2701 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → ∪ 𝑋 ∈ 𝑆) |
43 | 42 | ex 450 | . . . . . 6 ⊢ (𝜑 → (𝑓:ℕ–onto→𝑋 → ∪ 𝑋 ∈ 𝑆)) |
44 | 43 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (𝑓:ℕ–onto→𝑋 → ∪ 𝑋 ∈ 𝑆)) |
45 | 44 | exlimdv 1861 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (∃𝑓 𝑓:ℕ–onto→𝑋 → ∪ 𝑋 ∈ 𝑆)) |
46 | 30, 45 | mpd 15 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∪ 𝑋 ∈ 𝑆) |
47 | 10, 12, 46 | syl2anc 693 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∪ 𝑋 ∈ 𝑆) |
48 | 9, 47 | pm2.61dan 832 | 1 ⊢ (𝜑 → ∪ 𝑋 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ⊆ wss 3574 ∅c0 3915 ∪ cuni 4436 ∪ ciun 4520 class class class wbr 4653 Rel wrel 5119 ⟶wf 5884 –onto→wfo 5886 ‘cfv 5888 ωcom 7065 ≈ cen 7952 ≼ cdom 7953 ≺ csdm 7954 1c1 9937 ℕcn 11020 OutMeascome 40703 CaraGenccaragen 40705 |
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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-ac2 9285 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 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 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-rmo 2920 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-disj 4621 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-se 5074 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-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-ac 8939 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xadd 11947 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-sumge0 40580 df-ome 40704 df-caragen 40706 |
This theorem is referenced by: caragensal 40739 |
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