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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 7613 | . . 3 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V) | |
4 | uniexg 7466 | . . 3 ⊢ (dom 𝐹 ∈ V → ∪ dom 𝐹 ∈ V) | |
5 | 2, 3, 4 | 3syl 18 | . 2 ⊢ (𝜑 → ∪ dom 𝐹 ∈ V) |
6 | 1, 5 | eqeltrid 2917 | 1 ⊢ (𝜑 → 𝑋 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∪ cuni 4838 dom cdm 5555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-cnv 5563 df-dm 5565 df-rn 5566 |
This theorem is referenced by: omessle 42800 caragensplit 42802 omeunile 42807 caragenuncl 42815 omeunle 42818 omeiunlempt 42822 carageniuncllem2 42824 caragencmpl 42837 |
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