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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iunmapss | Structured version Visualization version GIF version |
Description: The indexed union of set exponentiations is a subset of the set exponentiation of the indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
iunmapss.x | ⊢ Ⅎ𝑥𝜑 |
iunmapss.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
iunmapss.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) |
Ref | Expression |
---|---|
iunmapss | ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐵 ↑m 𝐶) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunmapss.x | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | iunmapss.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | iunmapss.b | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) | |
4 | 3 | ex 412 | . . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑊)) |
5 | 1, 4 | ralrimi 3251 | . . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) |
6 | iunexg 7967 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) | |
7 | 2, 5, 6 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) |
8 | 7 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) |
9 | ssiun2 5050 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) | |
10 | 9 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
11 | mapss 8908 | . . . . 5 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ∈ V ∧ 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) → (𝐵 ↑m 𝐶) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐶)) | |
12 | 8, 10, 11 | syl2anc 583 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ↑m 𝐶) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐶)) |
13 | 12 | ex 412 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝐵 ↑m 𝐶) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐶))) |
14 | 1, 13 | ralrimi 3251 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐵 ↑m 𝐶) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐶)) |
15 | nfiu1 5030 | . . . 4 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵 | |
16 | nfcv 2899 | . . . 4 ⊢ Ⅎ𝑥 ↑m | |
17 | nfcv 2899 | . . . 4 ⊢ Ⅎ𝑥𝐶 | |
18 | 15, 16, 17 | nfov 7450 | . . 3 ⊢ Ⅎ𝑥(∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐶) |
19 | 18 | iunssf 5047 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 (𝐵 ↑m 𝐶) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐶) ↔ ∀𝑥 ∈ 𝐴 (𝐵 ↑m 𝐶) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐶)) |
20 | 14, 19 | sylibr 233 | 1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐵 ↑m 𝐶) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 Ⅎwnf 1778 ∈ wcel 2099 ∀wral 3058 Vcvv 3471 ⊆ wss 3947 ∪ ciun 4996 (class class class)co 7420 ↑m cmap 8845 |
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 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-map 8847 |
This theorem is referenced by: iunmapsn 44590 |
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