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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 4923 | . . . . 5 ⊢ (𝑋 = ∅ → ∪ 𝑋 = ∪ ∅) | |
2 | uni0 4940 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
3 | 1, 2 | eqtrdi 2791 | . . . 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 46462 | . . . 4 ⊢ (𝜑 → ∅ ∈ 𝑆) |
8 | 7 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ 𝑆) |
9 | 4, 8 | eqeltrd 2839 | . 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∪ 𝑋 ∈ 𝑆) |
10 | simpl 482 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝜑) | |
11 | neqne 2946 | . . . 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 8990 | . . . . . . . . . 10 ⊢ Rel ≼ | |
16 | brrelex1 5742 | . . . . . . . . . 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 9143 | . . . . . . 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 14018 | . . . . . . . . 9 ⊢ ℕ ≈ ω | |
24 | 23 | ensymi 9043 | . . . . . . . 8 ⊢ ω ≈ ℕ |
25 | 24 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ω ≈ ℕ) |
26 | domentr 9052 | . . . . . . 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 9167 | . . . . 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 45122 | . . . . . . . . 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 12646 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → 1 ∈ ℤ) | |
35 | nnuz 12919 | . . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1) | |
36 | fof 6821 | . . . . . . . . . . 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 6754 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → 𝑓:ℕ⟶𝑆) |
41 | 33, 6, 34, 35, 40 | carageniuncl 46479 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:ℕ–onto→𝑋) → ∪ 𝑛 ∈ ℕ (𝑓‘𝑛) ∈ 𝑆) |
42 | 32, 41 | eqeltrd 2839 | . . . . . . 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 1931 | . . . 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 813 | 1 ⊢ (𝜑 → ∪ 𝑋 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ⊆ wss 3963 ∅c0 4339 ∪ cuni 4912 ∪ ciun 4996 class class class wbr 5148 Rel wrel 5694 ⟶wf 6559 –onto→wfo 6561 ‘cfv 6563 ωcom 7887 ≈ cen 8981 ≼ cdom 8982 ≺ csdm 8983 1c1 11154 ℕcn 12264 OutMeascome 46445 CaraGenccaragen 46447 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-ac2 10501 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-acn 9980 df-ac 10154 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-xadd 13153 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-sumge0 46319 df-ome 46446 df-caragen 46448 |
This theorem is referenced by: caragensal 46481 |
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