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Mirrors > Home > MPE Home > Th. List > Mathboxes > djussxp2 | Structured version Visualization version GIF version |
Description: Stronger version of djussxp 5699. (Contributed by Thierry Arnoux, 23-Jun-2024.) |
Ref | Expression |
---|---|
djussxp2 | ⊢ ∪ 𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2897 | . . . 4 ⊢ Ⅎ𝑘𝐴 | |
2 | nfiu1 4924 | . . . 4 ⊢ Ⅎ𝑘∪ 𝑘 ∈ 𝐴 𝐵 | |
3 | 1, 2 | nfxp 5569 | . . 3 ⊢ Ⅎ𝑘(𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵) |
4 | 3 | iunssf 4939 | . 2 ⊢ (∪ 𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵) ↔ ∀𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵)) |
5 | snssi 4707 | . . 3 ⊢ (𝑘 ∈ 𝐴 → {𝑘} ⊆ 𝐴) | |
6 | ssiun2 4942 | . . 3 ⊢ (𝑘 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑘 ∈ 𝐴 𝐵) | |
7 | xpss12 5551 | . . 3 ⊢ (({𝑘} ⊆ 𝐴 ∧ 𝐵 ⊆ ∪ 𝑘 ∈ 𝐴 𝐵) → ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵)) | |
8 | 5, 6, 7 | syl2anc 587 | . 2 ⊢ (𝑘 ∈ 𝐴 → ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵)) |
9 | 4, 8 | mprgbir 3066 | 1 ⊢ ∪ 𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2112 ⊆ wss 3853 {csn 4527 ∪ ciun 4890 × cxp 5534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-v 3400 df-in 3860 df-ss 3870 df-sn 4528 df-iun 4892 df-opab 5102 df-xp 5542 |
This theorem is referenced by: 2ndresdju 30659 gsumpart 30988 |
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