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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 4917 | . . . . 5 ⊢ (𝑋 = ∅ → ∪ 𝑋 = ∪ ∅) | |
| 2 | uni0 4934 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
| 3 | 1, 2 | eqtrdi 2792 | . . . 4 ⊢ (𝑋 = ∅ → ∪ 𝑋 = ∅) | 
| 4 | 3 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑋 = ∅) | 
| 5 | caragenunicl.o | . . . . 5 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
| 6 | caragenunicl.s | . . . . 5 ⊢ 𝑆 = (CaraGen‘𝑂) | |
| 7 | 5, 6 | caragen0 46526 | . . . 4 ⊢ (𝜑 → ∅ ∈ 𝑆) | 
| 8 | 7 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ 𝑆) | 
| 9 | 4, 8 | eqeltrd 2840 | . 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑋 ∈ 𝑆) | 
| 10 | simpl 482 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝜑) | |
| 11 | neqne 2947 | . . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) | |
| 12 | 11 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅) | 
| 13 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ≠ ∅) | |
| 14 | caragenunicl.ctb | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≼ ω) | |
| 15 | reldom 8992 | . . . . . . . . . 10 ⊢ Rel ≼ | |
| 16 | brrelex1 5737 | . . . . . . . . . 10 ⊢ ((Rel ≼ ∧ 𝑋 ≼ ω) → 𝑋 ∈ V) | |
| 17 | 15, 16 | mpan 690 | . . . . . . . . 9 ⊢ (𝑋 ≼ ω → 𝑋 ∈ V) | 
| 18 | 14, 17 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ V) | 
| 19 | 18 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ∈ V) | 
| 20 | 0sdomg 9145 | . . . . . . 7 ⊢ (𝑋 ∈ V → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) | |
| 21 | 19, 20 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) | 
| 22 | 13, 21 | mpbird 257 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∅ ≺ 𝑋) | 
| 23 | nnenom 14022 | . . . . . . . . 9 ⊢ ℕ ≈ ω | |
| 24 | 23 | ensymi 9045 | . . . . . . . 8 ⊢ ω ≈ ℕ | 
| 25 | 24 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ω ≈ ℕ) | 
| 26 | domentr 9054 | . . . . . . 7 ⊢ ((𝑋 ≼ ω ∧ ω ≈ ℕ) → 𝑋 ≼ ℕ) | |
| 27 | 14, 25, 26 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → 𝑋 ≼ ℕ) | 
| 28 | 27 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ≼ ℕ) | 
| 29 | fodomr 9169 | . . . . 5 ⊢ ((∅ ≺ 𝑋 ∧ 𝑋 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto→𝑋) | |
| 30 | 22, 28, 29 | syl2anc 584 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∃𝑓 𝑓:ℕ–onto→𝑋) | 
| 31 | founiiun 45189 | . . . . . . . . 9 ⊢ (𝑓:ℕ–onto→𝑋 → ∪ 𝑋 = ∪ 𝑛 ∈ ℕ (𝑓‘𝑛)) | |
| 32 | 31 | adantl 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → ∪ 𝑋 = ∪ 𝑛 ∈ ℕ (𝑓‘𝑛)) | 
| 33 | 5 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → 𝑂 ∈ OutMeas) | 
| 34 | 1zzd 12650 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → 1 ∈ ℤ) | |
| 35 | nnuz 12922 | . . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1) | |
| 36 | fof 6819 | . . . . . . . . . . 11 ⊢ (𝑓:ℕ–onto→𝑋 → 𝑓:ℕ⟶𝑋) | |
| 37 | 36 | adantl 481 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → 𝑓:ℕ⟶𝑋) | 
| 38 | caragenunicl.y | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) | |
| 39 | 38 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → 𝑋 ⊆ 𝑆) | 
| 40 | 37, 39 | fssd 6752 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → 𝑓:ℕ⟶𝑆) | 
| 41 | 33, 6, 34, 35, 40 | carageniuncl 46543 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → ∪ 𝑛 ∈ ℕ (𝑓‘𝑛) ∈ 𝑆) | 
| 42 | 32, 41 | eqeltrd 2840 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → ∪ 𝑋 ∈ 𝑆) | 
| 43 | 42 | ex 412 | . . . . . 6 ⊢ (𝜑 → (𝑓:ℕ–onto→𝑋 → ∪ 𝑋 ∈ 𝑆)) | 
| 44 | 43 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (𝑓:ℕ–onto→𝑋 → ∪ 𝑋 ∈ 𝑆)) | 
| 45 | 44 | exlimdv 1932 | . . . 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 584 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∪ 𝑋 ∈ 𝑆) | 
| 48 | 9, 47 | pm2.61dan 812 | 1 ⊢ (𝜑 → ∪ 𝑋 ∈ 𝑆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ≠ wne 2939 Vcvv 3479 ⊆ wss 3950 ∅c0 4332 ∪ cuni 4906 ∪ ciun 4990 class class class wbr 5142 Rel wrel 5689 ⟶wf 6556 –onto→wfo 6558 ‘cfv 6560 ωcom 7888 ≈ cen 8983 ≼ cdom 8984 ≺ csdm 8985 1c1 11157 ℕcn 12267 OutMeascome 46509 CaraGenccaragen 46511 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-ac2 10504 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-disj 5110 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-omul 8512 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-acn 9983 df-ac 10157 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-n0 12529 df-z 12616 df-uz 12880 df-q 12992 df-rp 13036 df-xadd 13156 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-seq 14044 df-exp 14104 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-clim 15525 df-sum 15724 df-sumge0 46383 df-ome 46510 df-caragen 46512 | 
| This theorem is referenced by: caragensal 46545 | 
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