![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > iunctb | Structured version Visualization version GIF version |
Description: The countable union of countable sets is countable (indexed union version of unictb 10620). (Contributed by Mario Carneiro, 18-Jan-2014.) |
Ref | Expression |
---|---|
iunctb | ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) | |
2 | simpl 481 | . . . 4 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → 𝐴 ≼ ω) | |
3 | ctex 8996 | . . . . . . 7 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) | |
4 | 3 | adantr 479 | . . . . . 6 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → 𝐴 ∈ V) |
5 | ovex 7459 | . . . . . . 7 ⊢ (ω ↑m 𝐵) ∈ V | |
6 | 5 | rgenw 3055 | . . . . . 6 ⊢ ∀𝑥 ∈ 𝐴 (ω ↑m 𝐵) ∈ V |
7 | iunexg 7979 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 (ω ↑m 𝐵) ∈ V) → ∪ 𝑥 ∈ 𝐴 (ω ↑m 𝐵) ∈ V) | |
8 | 4, 6, 7 | sylancl 584 | . . . . 5 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → ∪ 𝑥 ∈ 𝐴 (ω ↑m 𝐵) ∈ V) |
9 | acncc 10485 | . . . . 5 ⊢ AC ω = V | |
10 | 8, 9 | eleqtrrdi 2837 | . . . 4 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → ∪ 𝑥 ∈ 𝐴 (ω ↑m 𝐵) ∈ AC ω) |
11 | acndom 10096 | . . . 4 ⊢ (𝐴 ≼ ω → (∪ 𝑥 ∈ 𝐴 (ω ↑m 𝐵) ∈ AC ω → ∪ 𝑥 ∈ 𝐴 (ω ↑m 𝐵) ∈ AC 𝐴)) | |
12 | 2, 10, 11 | sylc 65 | . . 3 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → ∪ 𝑥 ∈ 𝐴 (ω ↑m 𝐵) ∈ AC 𝐴) |
13 | simpr 483 | . . 3 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) | |
14 | omex 9688 | . . . . . 6 ⊢ ω ∈ V | |
15 | xpdom1g 9109 | . . . . . 6 ⊢ ((ω ∈ V ∧ 𝐴 ≼ ω) → (𝐴 × ω) ≼ (ω × ω)) | |
16 | 14, 2, 15 | sylancr 585 | . . . . 5 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → (𝐴 × ω) ≼ (ω × ω)) |
17 | xpomen 10060 | . . . . 5 ⊢ (ω × ω) ≈ ω | |
18 | domentr 9046 | . . . . 5 ⊢ (((𝐴 × ω) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝐴 × ω) ≼ ω) | |
19 | 16, 17, 18 | sylancl 584 | . . . 4 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → (𝐴 × ω) ≼ ω) |
20 | ctex 8996 | . . . . . . 7 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V) | |
21 | 20 | ralimi 3073 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≼ ω → ∀𝑥 ∈ 𝐴 𝐵 ∈ V) |
22 | iunexg 7979 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ V) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) | |
23 | 3, 21, 22 | syl2an 594 | . . . . 5 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) |
24 | omelon 9691 | . . . . . 6 ⊢ ω ∈ On | |
25 | onenon 9994 | . . . . . 6 ⊢ (ω ∈ On → ω ∈ dom card) | |
26 | 24, 25 | ax-mp 5 | . . . . 5 ⊢ ω ∈ dom card |
27 | numacn 10094 | . . . . 5 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ V → (ω ∈ dom card → ω ∈ AC ∪ 𝑥 ∈ 𝐴 𝐵)) | |
28 | 23, 26, 27 | mpisyl 21 | . . . 4 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → ω ∈ AC ∪ 𝑥 ∈ 𝐴 𝐵) |
29 | acndom2 10099 | . . . 4 ⊢ ((𝐴 × ω) ≼ ω → (ω ∈ AC ∪ 𝑥 ∈ 𝐴 𝐵 → (𝐴 × ω) ∈ AC ∪ 𝑥 ∈ 𝐴 𝐵)) | |
30 | 19, 28, 29 | sylc 65 | . . 3 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → (𝐴 × ω) ∈ AC ∪ 𝑥 ∈ 𝐴 𝐵) |
31 | 1, 12, 13, 30 | iundomg 10586 | . 2 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ (𝐴 × ω)) |
32 | domtr 9040 | . 2 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ≼ (𝐴 × ω) ∧ (𝐴 × ω) ≼ ω) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ ω) | |
33 | 31, 19, 32 | syl2anc 582 | 1 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2099 ∀wral 3051 Vcvv 3462 {csn 4633 ∪ ciun 5003 class class class wbr 5155 × cxp 5682 dom cdm 5684 Oncon0 6378 (class class class)co 7426 ωcom 7878 ↑m cmap 8857 ≈ cen 8973 ≼ cdom 8974 cardccrd 9980 AC wacn 9983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-inf2 9686 ax-cc 10480 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-int 4957 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-se 5640 df-we 5641 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 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-isom 6565 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8005 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-er 8736 df-map 8859 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-oi 9555 df-card 9984 df-acn 9987 |
This theorem is referenced by: unictb 10620 iunctb2 37112 heiborlem3 37516 |
Copyright terms: Public domain | W3C validator |