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| Mirrors > Home > MPE Home > Th. List > Mathboxes > unidmex | Structured version Visualization version GIF version | ||
| Description: If 𝐹 is a set, then ∪ dom 𝐹 is a set. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
| Ref | Expression |
|---|---|
| unidmex.f | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
| unidmex.x | ⊢ 𝑋 = ∪ dom 𝐹 |
| Ref | Expression |
|---|---|
| unidmex | ⊢ (𝜑 → 𝑋 ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | unidmex.x | . 2 ⊢ 𝑋 = ∪ dom 𝐹 | |
| 2 | unidmex.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
| 3 | dmexg 7923 | . . 3 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V) | |
| 4 | uniexg 7760 | . . 3 ⊢ (dom 𝐹 ∈ V → ∪ dom 𝐹 ∈ V) | |
| 5 | 2, 3, 4 | 3syl 18 | . 2 ⊢ (𝜑 → ∪ dom 𝐹 ∈ V) |
| 6 | 1, 5 | eqeltrid 2845 | 1 ⊢ (𝜑 → 𝑋 ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cuni 4907 dom cdm 5685 |
| 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-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 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-uni 4908 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: omessle 46513 caragensplit 46515 omeunile 46520 caragenuncl 46528 omeunle 46531 omeiunlempt 46535 carageniuncllem2 46537 caragencmpl 46550 |
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