| 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 2920 | . . . 4 ⊢ Ⅎ𝑥 𝐶 ∈ 𝐴 |
| 4 | ssiun2sf.3 | . . . . 5 ⊢ Ⅎ𝑥𝐷 | |
| 5 | nfiu1 5027 | . . . . 5 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵 | |
| 6 | 4, 5 | nfss 3976 | . . . 4 ⊢ Ⅎ𝑥 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 |
| 7 | 3, 6 | nfim 1896 | . . 3 ⊢ Ⅎ𝑥(𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
| 8 | eleq1 2829 | . . . 4 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
| 9 | ssiun2sf.4 | . . . . 5 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) | |
| 10 | 9 | sseq1d 4015 | . . . 4 ⊢ (𝑥 = 𝐶 → (𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)) |
| 11 | 8, 10 | imbi12d 344 | . . 3 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))) |
| 12 | ssiun2 5047 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) | |
| 13 | 1, 7, 11, 12 | vtoclgf 3569 | . 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 1540 ∈ wcel 2108 Ⅎwnfc 2890 ⊆ wss 3951 ∪ ciun 4991 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-v 3482 df-ss 3968 df-iun 4993 |
| This theorem is referenced by: iundisj2f 32603 esum2dlem 34093 voliune 34230 volfiniune 34231 |
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