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 415 | . . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑊)) |
5 | 1, 4 | ralrimi 3218 | . . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) |
6 | iunexg 7666 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) | |
7 | 2, 5, 6 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) |
8 | 7 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ V) |
9 | ssiun2 4973 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) | |
10 | 9 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
11 | mapss 8455 | . . . . 5 ⊢ ((∪ 𝑥 ∈ 𝐴 𝐵 ∈ V ∧ 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) → (𝐵 ↑m 𝐶) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐶)) | |
12 | 8, 10, 11 | syl2anc 586 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ↑m 𝐶) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐶)) |
13 | 12 | ex 415 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝐵 ↑m 𝐶) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐶))) |
14 | 1, 13 | ralrimi 3218 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐵 ↑m 𝐶) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐶)) |
15 | nfiu1 4955 | . . . 4 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵 | |
16 | nfcv 2979 | . . . 4 ⊢ Ⅎ𝑥 ↑m | |
17 | nfcv 2979 | . . . 4 ⊢ Ⅎ𝑥𝐶 | |
18 | 15, 16, 17 | nfov 7188 | . . 3 ⊢ Ⅎ𝑥(∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐶) |
19 | 18 | iunssf 41359 | . 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 236 | 1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐵 ↑m 𝐶) ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 Ⅎwnf 1784 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 ⊆ wss 3938 ∪ ciun 4921 (class class class)co 7158 ↑m cmap 8408 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 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-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-map 8410 |
This theorem is referenced by: iunmapsn 41487 |
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