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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 3484 | . . . 4 ⊢ 𝑥 ∈ V | |
| 2 | vex 3484 | . . . 4 ⊢ 𝑦 ∈ V | |
| 3 | 1, 2 | elimasn 6108 | . . 3 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑥}) ↔ 〈𝑥, 𝑦〉 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)) |
| 4 | opeliunxp 5752 | . . . . 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 2735 | 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 1540 ∈ wcel 2108 {csn 4626 〈cop 4632 ∪ ciun 4991 × cxp 5683 “ cima 5688 |
| 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 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 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-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-iun 4993 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 |
| This theorem is referenced by: esum2d 34094 |
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