Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssiun2sf | Structured version Visualization version GIF version |
Description: Subset relationship for an indexed union. (Contributed by Thierry Arnoux, 31-Dec-2016.) |
Ref | Expression |
---|---|
ssiun2sf.1 | ⊢ Ⅎ𝑥𝐴 |
ssiun2sf.2 | ⊢ Ⅎ𝑥𝐶 |
ssiun2sf.3 | ⊢ Ⅎ𝑥𝐷 |
ssiun2sf.4 | ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
ssiun2sf | ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssiun2sf.2 | . . 3 ⊢ Ⅎ𝑥𝐶 | |
2 | ssiun2sf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 1, 2 | nfel 2921 | . . . 4 ⊢ Ⅎ𝑥 𝐶 ∈ 𝐴 |
4 | ssiun2sf.3 | . . . . 5 ⊢ Ⅎ𝑥𝐷 | |
5 | nfiu1 4958 | . . . . 5 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵 | |
6 | 4, 5 | nfss 3913 | . . . 4 ⊢ Ⅎ𝑥 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 |
7 | 3, 6 | nfim 1899 | . . 3 ⊢ Ⅎ𝑥(𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
8 | eleq1 2826 | . . . 4 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
9 | ssiun2sf.4 | . . . . 5 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) | |
10 | 9 | sseq1d 3952 | . . . 4 ⊢ (𝑥 = 𝐶 → (𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)) |
11 | 8, 10 | imbi12d 345 | . . 3 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))) |
12 | ssiun2 4977 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) | |
13 | 1, 7, 11, 12 | vtoclgf 3503 | . 2 ⊢ (𝐶 ∈ 𝐴 → (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)) |
14 | 13 | pm2.43i 52 | 1 ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 Ⅎwnfc 2887 ⊆ wss 3887 ∪ ciun 4924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-v 3434 df-in 3894 df-ss 3904 df-iun 4926 |
This theorem is referenced by: iundisj2f 30929 esum2dlem 32060 voliune 32197 volfiniune 32198 |
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