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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 (common case). (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 6989 | . . 3 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V) | |
4 | uniexg 6853 | . . 3 ⊢ (dom 𝐹 ∈ V → ∪ dom 𝐹 ∈ V) | |
5 | 2, 3, 4 | 3syl 18 | . 2 ⊢ (𝜑 → ∪ dom 𝐹 ∈ V) |
6 | 1, 5 | syl5eqel 2692 | 1 ⊢ (𝜑 → 𝑋 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cuni 4372 dom cdm 5038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: omessle 39388 caragensplit 39390 omeunile 39395 caragenuncl 39403 omeunle 39406 omeiunlempt 39410 carageniuncllem2 39412 caragencmpl 39425 |
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