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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 411 | . . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑊)) |
5 | 1, 4 | ralrimi 3252 | . . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) |
6 | iunexg 7973 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) | |
7 | 2, 5, 6 | syl2anc 582 | . . . . . 6 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) |
8 | 7 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) |
9 | ssiun2 5054 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) | |
10 | 9 | adantl 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
11 | mapss 8914 | . . . . 5 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ∈ V ∧ 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) → (𝐵 ↑m 𝐶) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐶)) | |
12 | 8, 10, 11 | syl2anc 582 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ↑m 𝐶) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐶)) |
13 | 12 | ex 411 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝐵 ↑m 𝐶) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐶))) |
14 | 1, 13 | ralrimi 3252 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐵 ↑m 𝐶) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐶)) |
15 | nfiu1 5034 | . . . 4 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵 | |
16 | nfcv 2899 | . . . 4 ⊢ Ⅎ𝑥 ↑m | |
17 | nfcv 2899 | . . . 4 ⊢ Ⅎ𝑥𝐶 | |
18 | 15, 16, 17 | nfov 7456 | . . 3 ⊢ Ⅎ𝑥(∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐶) |
19 | 18 | iunssf 5051 | . 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 394 Ⅎwnf 1777 ∈ wcel 2098 ∀wral 3058 Vcvv 3473 ⊆ wss 3949 ∪ ciun 5000 (class class class)co 7426 ↑m cmap 8851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-map 8853 |
This theorem is referenced by: iunmapsn 44620 |
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