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Mirrors > Home > ILE Home > Th. List > ssiun2s | GIF version |
Description: Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.) |
Ref | Expression |
---|---|
ssiun2s.1 | ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
ssiun2s | ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2219 | . 2 ⊢ Ⅎ𝑥𝐶 | |
2 | nfcv 2219 | . . 3 ⊢ Ⅎ𝑥𝐷 | |
3 | nfiu1 3708 | . . 3 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵 | |
4 | 2, 3 | nfss 2992 | . 2 ⊢ Ⅎ𝑥 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 |
5 | ssiun2s.1 | . . 3 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) | |
6 | 5 | sseq1d 3026 | . 2 ⊢ (𝑥 = 𝐶 → (𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)) |
7 | ssiun2 3721 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) | |
8 | 1, 4, 6, 7 | vtoclgaf 2663 | 1 ⊢ (𝐶 ∈ 𝐴 → 𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ∈ wcel 1433 ⊆ wss 2973 ∪ ciun 3678 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-in 2979 df-ss 2986 df-iun 3680 |
This theorem is referenced by: (None) |
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