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Mirrors > Home > MPE Home > Th. List > iundom | Structured version Visualization version GIF version |
Description: An upper bound for the cardinality of an indexed union. 𝐶 depends on 𝑥 and should be thought of as 𝐶(𝑥). (Contributed by NM, 26-Mar-2006.) |
Ref | Expression |
---|---|
iundom | ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐶 ≼ (𝐴 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐶) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐶) | |
2 | simpl 482 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵) → 𝐴 ∈ 𝑉) | |
3 | ovex 7463 | . . . . . 6 ⊢ (𝐵 ↑m 𝐶) ∈ V | |
4 | 3 | rgenw 3062 | . . . . 5 ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ↑m 𝐶) ∈ V |
5 | iunexg 7986 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (𝐵 ↑m 𝐶) ∈ V) → ∪ 𝑥 ∈ 𝐴 (𝐵 ↑m 𝐶) ∈ V) | |
6 | 2, 4, 5 | sylancl 586 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵) → ∪ 𝑥 ∈ 𝐴 (𝐵 ↑m 𝐶) ∈ V) |
7 | numth3 10507 | . . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 (𝐵 ↑m 𝐶) ∈ V → ∪ 𝑥 ∈ 𝐴 (𝐵 ↑m 𝐶) ∈ dom card) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵) → ∪ 𝑥 ∈ 𝐴 (𝐵 ↑m 𝐶) ∈ dom card) |
9 | numacn 10086 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝑥 ∈ 𝐴 (𝐵 ↑m 𝐶) ∈ dom card → ∪ 𝑥 ∈ 𝐴 (𝐵 ↑m 𝐶) ∈ AC 𝐴)) | |
10 | 2, 8, 9 | sylc 65 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵) → ∪ 𝑥 ∈ 𝐴 (𝐵 ↑m 𝐶) ∈ AC 𝐴) |
11 | simpr 484 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵) → ∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵) | |
12 | reldom 8989 | . . . . . 6 ⊢ Rel ≼ | |
13 | 12 | brrelex1i 5744 | . . . . 5 ⊢ (𝐶 ≼ 𝐵 → 𝐶 ∈ V) |
14 | 13 | ralimi 3080 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵 → ∀𝑥 ∈ 𝐴 𝐶 ∈ V) |
15 | iunexg 7986 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ V) → ∪ 𝑥 ∈ 𝐴 𝐶 ∈ V) | |
16 | 14, 15 | sylan2 593 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐶 ∈ V) |
17 | 1, 10, 11 | iundom2g 10577 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵) → ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐶) ≼ (𝐴 × 𝐵)) |
18 | 12 | brrelex2i 5745 | . . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐶) ≼ (𝐴 × 𝐵) → (𝐴 × 𝐵) ∈ V) |
19 | numth3 10507 | . . . 4 ⊢ ((𝐴 × 𝐵) ∈ V → (𝐴 × 𝐵) ∈ dom card) | |
20 | 17, 18, 19 | 3syl 18 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵) → (𝐴 × 𝐵) ∈ dom card) |
21 | numacn 10086 | . . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝐶 ∈ V → ((𝐴 × 𝐵) ∈ dom card → (𝐴 × 𝐵) ∈ AC ∪ 𝑥 ∈ 𝐴 𝐶)) | |
22 | 16, 20, 21 | sylc 65 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵) → (𝐴 × 𝐵) ∈ AC ∪ 𝑥 ∈ 𝐴 𝐶) |
23 | 1, 10, 11, 22 | iundomg 10578 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐶 ≼ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 ∀wral 3058 Vcvv 3477 {csn 4630 ∪ ciun 4995 class class class wbr 5147 × cxp 5686 dom cdm 5688 (class class class)co 7430 ↑m cmap 8864 ≼ cdom 8981 cardccrd 9972 AC wacn 9975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-ac2 10500 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-card 9976 df-acn 9979 df-ac 10153 |
This theorem is referenced by: unidom 10580 alephreg 10619 inar1 10812 |
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