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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 4716 | . . . . 5 ⊢ (𝑋 = ∅ → ∪ 𝑋 = ∪ ∅) | |
2 | uni0 4735 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
3 | 1, 2 | syl6eq 2824 | . . . 4 ⊢ (𝑋 = ∅ → ∪ 𝑋 = ∅) |
4 | 3 | adantl 474 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑋 = ∅) |
5 | caragenunicl.o | . . . . 5 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
6 | caragenunicl.s | . . . . 5 ⊢ 𝑆 = (CaraGen‘𝑂) | |
7 | 5, 6 | caragen0 42244 | . . . 4 ⊢ (𝜑 → ∅ ∈ 𝑆) |
8 | 7 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ 𝑆) |
9 | 4, 8 | eqeltrd 2860 | . 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑋 ∈ 𝑆) |
10 | simpl 475 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝜑) | |
11 | neqne 2969 | . . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) | |
12 | 11 | adantl 474 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅) |
13 | simpr 477 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ≠ ∅) | |
14 | caragenunicl.ctb | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≼ ω) | |
15 | reldom 8310 | . . . . . . . . . 10 ⊢ Rel ≼ | |
16 | brrelex1 5451 | . . . . . . . . . 10 ⊢ ((Rel ≼ ∧ 𝑋 ≼ ω) → 𝑋 ∈ V) | |
17 | 15, 16 | mpan 677 | . . . . . . . . 9 ⊢ (𝑋 ≼ ω → 𝑋 ∈ V) |
18 | 14, 17 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ V) |
19 | 18 | adantr 473 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ∈ V) |
20 | 0sdomg 8440 | . . . . . . 7 ⊢ (𝑋 ∈ V → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) | |
21 | 19, 20 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) |
22 | 13, 21 | mpbird 249 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∅ ≺ 𝑋) |
23 | nnenom 13161 | . . . . . . . . 9 ⊢ ℕ ≈ ω | |
24 | 23 | ensymi 8354 | . . . . . . . 8 ⊢ ω ≈ ℕ |
25 | 24 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ω ≈ ℕ) |
26 | domentr 8363 | . . . . . . 7 ⊢ ((𝑋 ≼ ω ∧ ω ≈ ℕ) → 𝑋 ≼ ℕ) | |
27 | 14, 25, 26 | syl2anc 576 | . . . . . 6 ⊢ (𝜑 → 𝑋 ≼ ℕ) |
28 | 27 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ≼ ℕ) |
29 | fodomr 8462 | . . . . 5 ⊢ ((∅ ≺ 𝑋 ∧ 𝑋 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto→𝑋) | |
30 | 22, 28, 29 | syl2anc 576 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∃𝑓 𝑓:ℕ–onto→𝑋) |
31 | founiiun 40885 | . . . . . . . . 9 ⊢ (𝑓:ℕ–onto→𝑋 → ∪ 𝑋 = ∪ 𝑛 ∈ ℕ (𝑓‘𝑛)) | |
32 | 31 | adantl 474 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → ∪ 𝑋 = ∪ 𝑛 ∈ ℕ (𝑓‘𝑛)) |
33 | 5 | adantr 473 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → 𝑂 ∈ OutMeas) |
34 | 1zzd 11824 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → 1 ∈ ℤ) | |
35 | nnuz 12093 | . . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1) | |
36 | fof 6416 | . . . . . . . . . . 11 ⊢ (𝑓:ℕ–onto→𝑋 → 𝑓:ℕ⟶𝑋) | |
37 | 36 | adantl 474 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → 𝑓:ℕ⟶𝑋) |
38 | caragenunicl.y | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) | |
39 | 38 | adantr 473 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → 𝑋 ⊆ 𝑆) |
40 | 37, 39 | fssd 6355 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → 𝑓:ℕ⟶𝑆) |
41 | 33, 6, 34, 35, 40 | carageniuncl 42261 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → ∪ 𝑛 ∈ ℕ (𝑓‘𝑛) ∈ 𝑆) |
42 | 32, 41 | eqeltrd 2860 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → ∪ 𝑋 ∈ 𝑆) |
43 | 42 | ex 405 | . . . . . 6 ⊢ (𝜑 → (𝑓:ℕ–onto→𝑋 → ∪ 𝑋 ∈ 𝑆)) |
44 | 43 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (𝑓:ℕ–onto→𝑋 → ∪ 𝑋 ∈ 𝑆)) |
45 | 44 | exlimdv 1892 | . . . 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 576 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∪ 𝑋 ∈ 𝑆) |
48 | 9, 47 | pm2.61dan 800 | 1 ⊢ (𝜑 → ∪ 𝑋 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∃wex 1742 ∈ wcel 2050 ≠ wne 2961 Vcvv 3409 ⊆ wss 3823 ∅c0 4172 ∪ cuni 4708 ∪ ciun 4788 class class class wbr 4925 Rel wrel 5408 ⟶wf 6181 –onto→wfo 6183 ‘cfv 6185 ωcom 7394 ≈ cen 8301 ≼ cdom 8302 ≺ csdm 8303 1c1 10334 ℕcn 11437 OutMeascome 42227 CaraGenccaragen 42229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-inf2 8896 ax-ac2 9681 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-disj 4894 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-omul 7908 df-er 8087 df-map 8206 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-sup 8699 df-inf 8700 df-oi 8767 df-card 9160 df-acn 9163 df-ac 9334 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-3 11502 df-n0 11706 df-z 11792 df-uz 12057 df-q 12161 df-rp 12203 df-xadd 12323 df-ico 12558 df-icc 12559 df-fz 12707 df-fzo 12848 df-seq 13183 df-exp 13243 df-hash 13504 df-cj 14317 df-re 14318 df-im 14319 df-sqrt 14453 df-abs 14454 df-clim 14704 df-sum 14902 df-sumge0 42101 df-ome 42228 df-caragen 42230 |
This theorem is referenced by: caragensal 42263 |
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