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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iunsnima | Structured version Visualization version GIF version |
Description: Image of a singleton by an indexed union involving that singleton. (Contributed by Thierry Arnoux, 10-Apr-2020.) |
Ref | Expression |
---|---|
iunsnima.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
iunsnima.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) |
Ref | Expression |
---|---|
iunsnima | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑥}) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3481 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | vex 3481 | . . . 4 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | elimasn 6109 | . . 3 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑥}) ↔ 〈𝑥, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) |
4 | opeliunxp 5755 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
5 | 4 | baib 535 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (〈𝑥, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ 𝑦 ∈ 𝐵)) |
6 | 5 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (〈𝑥, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ 𝑦 ∈ 𝐵)) |
7 | 3, 6 | bitrid 283 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑥}) ↔ 𝑦 ∈ 𝐵)) |
8 | 7 | eqrdv 2732 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑥}) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 {csn 4630 〈cop 4636 ∪ ciun 4995 × cxp 5686 “ cima 5691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-iun 4997 df-br 5148 df-opab 5210 df-xp 5694 df-cnv 5696 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 |
This theorem is referenced by: esum2d 34073 |
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