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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 9999). (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 2823 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) | |
2 | simpl 485 | . . . 4 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → 𝐴 ≼ ω) | |
3 | ctex 8526 | . . . . . . 7 ⊢ (𝐴 ≼ ω → 𝐴 ∈ V) | |
4 | 3 | adantr 483 | . . . . . 6 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → 𝐴 ∈ V) |
5 | ovex 7191 | . . . . . . 7 ⊢ (ω ↑m 𝐵) ∈ V | |
6 | 5 | rgenw 3152 | . . . . . 6 ⊢ ∀𝑥 ∈ 𝐴 (ω ↑m 𝐵) ∈ V |
7 | iunexg 7666 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 (ω ↑m 𝐵) ∈ V) → ∪ 𝑥 ∈ 𝐴 (ω ↑m 𝐵) ∈ V) | |
8 | 4, 6, 7 | sylancl 588 | . . . . 5 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → ∪ 𝑥 ∈ 𝐴 (ω ↑m 𝐵) ∈ V) |
9 | acncc 9864 | . . . . 5 ⊢ AC ω = V | |
10 | 8, 9 | eleqtrrdi 2926 | . . . 4 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → ∪ 𝑥 ∈ 𝐴 (ω ↑m 𝐵) ∈ AC ω) |
11 | acndom 9479 | . . . 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 487 | . . 3 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) | |
14 | omex 9108 | . . . . . 6 ⊢ ω ∈ V | |
15 | xpdom1g 8616 | . . . . . 6 ⊢ ((ω ∈ V ∧ 𝐴 ≼ ω) → (𝐴 × ω) ≼ (ω × ω)) | |
16 | 14, 2, 15 | sylancr 589 | . . . . 5 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → (𝐴 × ω) ≼ (ω × ω)) |
17 | xpomen 9443 | . . . . 5 ⊢ (ω × ω) ≈ ω | |
18 | domentr 8570 | . . . . 5 ⊢ (((𝐴 × ω) ≼ (ω × ω) ∧ (ω × ω) ≈ ω) → (𝐴 × ω) ≼ ω) | |
19 | 16, 17, 18 | sylancl 588 | . . . 4 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → (𝐴 × ω) ≼ ω) |
20 | ctex 8526 | . . . . . . 7 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V) | |
21 | 20 | ralimi 3162 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≼ ω → ∀𝑥 ∈ 𝐴 𝐵 ∈ V) |
22 | iunexg 7666 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ V) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) | |
23 | 3, 21, 22 | syl2an 597 | . . . . 5 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) |
24 | omelon 9111 | . . . . . 6 ⊢ ω ∈ On | |
25 | onenon 9380 | . . . . . 6 ⊢ (ω ∈ On → ω ∈ dom card) | |
26 | 24, 25 | ax-mp 5 | . . . . 5 ⊢ ω ∈ dom card |
27 | numacn 9477 | . . . . 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 9482 | . . . 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 9965 | . 2 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ (𝐴 × ω)) |
32 | domtr 8564 | . 2 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ≼ (𝐴 × ω) ∧ (𝐴 × ω) ≼ ω) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ ω) | |
33 | 31, 19, 32 | syl2anc 586 | 1 ⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 {csn 4569 ∪ ciun 4921 class class class wbr 5068 × cxp 5555 dom cdm 5557 Oncon0 6193 (class class class)co 7158 ωcom 7582 ↑m cmap 8408 ≈ cen 8508 ≼ cdom 8509 cardccrd 9366 AC wacn 9369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cc 9859 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-oi 8976 df-card 9370 df-acn 9373 |
This theorem is referenced by: unictb 9999 iunctb2 34686 heiborlem3 35093 |
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