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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 2777 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐶) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐶) | |
2 | simpl 476 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵) → 𝐴 ∈ 𝑉) | |
3 | ovex 6954 | . . . . . 6 ⊢ (𝐵 ↑𝑚 𝐶) ∈ V | |
4 | 3 | rgenw 3105 | . . . . 5 ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ↑𝑚 𝐶) ∈ V |
5 | iunexg 7421 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (𝐵 ↑𝑚 𝐶) ∈ V) → ∪ 𝑥 ∈ 𝐴 (𝐵 ↑𝑚 𝐶) ∈ V) | |
6 | 2, 4, 5 | sylancl 580 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵) → ∪ 𝑥 ∈ 𝐴 (𝐵 ↑𝑚 𝐶) ∈ V) |
7 | numth3 9627 | . . . 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 9205 | . . 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 479 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵) → ∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵) | |
12 | reldom 8247 | . . . . . 6 ⊢ Rel ≼ | |
13 | 12 | brrelex1i 5406 | . . . . 5 ⊢ (𝐶 ≼ 𝐵 → 𝐶 ∈ V) |
14 | 13 | ralimi 3133 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵 → ∀𝑥 ∈ 𝐴 𝐶 ∈ V) |
15 | iunexg 7421 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ V) → ∪ 𝑥 ∈ 𝐴 𝐶 ∈ V) | |
16 | 14, 15 | sylan2 586 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐶 ∈ V) |
17 | 1, 10, 11 | iundom2g 9697 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵) → ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐶) ≼ (𝐴 × 𝐵)) |
18 | 12 | brrelex2i 5407 | . . . 4 ⊢ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐶) ≼ (𝐴 × 𝐵) → (𝐴 × 𝐵) ∈ V) |
19 | numth3 9627 | . . . 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 9205 | . . 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 9698 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐶 ≼ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2106 ∀wral 3089 Vcvv 3397 {csn 4397 ∪ ciun 4753 class class class wbr 4886 × cxp 5353 dom cdm 5355 (class class class)co 6922 ↑𝑚 cmap 8140 ≼ cdom 8239 cardccrd 9094 AC wacn 9097 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-ac2 9620 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-card 9098 df-acn 9101 df-ac 9272 |
This theorem is referenced by: unidom 9700 alephreg 9739 inar1 9932 |
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