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 4840 | . . . . 5 ⊢ (𝑋 = ∅ → ∪ 𝑋 = ∪ ∅) | |
2 | uni0 4859 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
3 | 1, 2 | syl6eq 2872 | . . . 4 ⊢ (𝑋 = ∅ → ∪ 𝑋 = ∅) |
4 | 3 | adantl 484 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑋 = ∅) |
5 | caragenunicl.o | . . . . 5 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
6 | caragenunicl.s | . . . . 5 ⊢ 𝑆 = (CaraGen‘𝑂) | |
7 | 5, 6 | caragen0 42781 | . . . 4 ⊢ (𝜑 → ∅ ∈ 𝑆) |
8 | 7 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ 𝑆) |
9 | 4, 8 | eqeltrd 2913 | . 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑋 ∈ 𝑆) |
10 | simpl 485 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝜑) | |
11 | neqne 3024 | . . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅) | |
12 | 11 | adantl 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅) |
13 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ≠ ∅) | |
14 | caragenunicl.ctb | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≼ ω) | |
15 | reldom 8509 | . . . . . . . . . 10 ⊢ Rel ≼ | |
16 | brrelex1 5600 | . . . . . . . . . 10 ⊢ ((Rel ≼ ∧ 𝑋 ≼ ω) → 𝑋 ∈ V) | |
17 | 15, 16 | mpan 688 | . . . . . . . . 9 ⊢ (𝑋 ≼ ω → 𝑋 ∈ V) |
18 | 14, 17 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ V) |
19 | 18 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ∈ V) |
20 | 0sdomg 8640 | . . . . . . 7 ⊢ (𝑋 ∈ V → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) | |
21 | 19, 20 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅)) |
22 | 13, 21 | mpbird 259 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∅ ≺ 𝑋) |
23 | nnenom 13342 | . . . . . . . . 9 ⊢ ℕ ≈ ω | |
24 | 23 | ensymi 8553 | . . . . . . . 8 ⊢ ω ≈ ℕ |
25 | 24 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ω ≈ ℕ) |
26 | domentr 8562 | . . . . . . 7 ⊢ ((𝑋 ≼ ω ∧ ω ≈ ℕ) → 𝑋 ≼ ℕ) | |
27 | 14, 25, 26 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → 𝑋 ≼ ℕ) |
28 | 27 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ≼ ℕ) |
29 | fodomr 8662 | . . . . 5 ⊢ ((∅ ≺ 𝑋 ∧ 𝑋 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto→𝑋) | |
30 | 22, 28, 29 | syl2anc 586 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ∃𝑓 𝑓:ℕ–onto→𝑋) |
31 | founiiun 41427 | . . . . . . . . 9 ⊢ (𝑓:ℕ–onto→𝑋 → ∪ 𝑋 = ∪ 𝑛 ∈ ℕ (𝑓‘𝑛)) | |
32 | 31 | adantl 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → ∪ 𝑋 = ∪ 𝑛 ∈ ℕ (𝑓‘𝑛)) |
33 | 5 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → 𝑂 ∈ OutMeas) |
34 | 1zzd 12007 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → 1 ∈ ℤ) | |
35 | nnuz 12275 | . . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1) | |
36 | fof 6585 | . . . . . . . . . . 11 ⊢ (𝑓:ℕ–onto→𝑋 → 𝑓:ℕ⟶𝑋) | |
37 | 36 | adantl 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → 𝑓:ℕ⟶𝑋) |
38 | caragenunicl.y | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) | |
39 | 38 | adantr 483 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → 𝑋 ⊆ 𝑆) |
40 | 37, 39 | fssd 6523 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → 𝑓:ℕ⟶𝑆) |
41 | 33, 6, 34, 35, 40 | carageniuncl 42798 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → ∪ 𝑛 ∈ ℕ (𝑓‘𝑛) ∈ 𝑆) |
42 | 32, 41 | eqeltrd 2913 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → ∪ 𝑋 ∈ 𝑆) |
43 | 42 | ex 415 | . . . . . 6 ⊢ (𝜑 → (𝑓:ℕ–onto→𝑋 → ∪ 𝑋 ∈ 𝑆)) |
44 | 43 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (𝑓:ℕ–onto→𝑋 → ∪ 𝑋 ∈ 𝑆)) |
45 | 44 | exlimdv 1930 | . . . 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 586 | . 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∪ 𝑋 ∈ 𝑆) |
48 | 9, 47 | pm2.61dan 811 | 1 ⊢ (𝜑 → ∪ 𝑋 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ≠ wne 3016 Vcvv 3495 ⊆ wss 3936 ∅c0 4291 ∪ cuni 4832 ∪ ciun 4912 class class class wbr 5059 Rel wrel 5555 ⟶wf 6346 –onto→wfo 6348 ‘cfv 6350 ωcom 7574 ≈ cen 8500 ≼ cdom 8501 ≺ csdm 8502 1c1 10532 ℕcn 11632 OutMeascome 42764 CaraGenccaragen 42766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-ac2 9879 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-disj 5025 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-omul 8101 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-acn 9365 df-ac 9536 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xadd 12502 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-sumge0 42638 df-ome 42765 df-caragen 42767 |
This theorem is referenced by: caragensal 42800 |
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