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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 2906 | . . . 4 ⊢ Ⅎ𝑥 𝐶 ∈ 𝐴 |
4 | ssiun2sf.3 | . . . . 5 ⊢ Ⅎ𝑥𝐷 | |
5 | nfiu1 5031 | . . . . 5 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵 | |
6 | 4, 5 | nfss 3969 | . . . 4 ⊢ Ⅎ𝑥 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 |
7 | 3, 6 | nfim 1891 | . . 3 ⊢ Ⅎ𝑥(𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
8 | eleq1 2813 | . . . 4 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
9 | ssiun2sf.4 | . . . . 5 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) | |
10 | 9 | sseq1d 4008 | . . . 4 ⊢ (𝑥 = 𝐶 → (𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)) |
11 | 8, 10 | imbi12d 343 | . . 3 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))) |
12 | ssiun2 5051 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) | |
13 | 1, 7, 11, 12 | vtoclgf 3548 | . 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 1533 ∈ wcel 2098 Ⅎwnfc 2875 ⊆ wss 3944 ∪ ciun 4997 |
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 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-v 3463 df-ss 3961 df-iun 4999 |
This theorem is referenced by: iundisj2f 32459 esum2dlem 33842 voliune 33979 volfiniune 33980 |
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