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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 7097 | . . 3 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V) | |
4 | uniexg 6955 | . . 3 ⊢ (dom 𝐹 ∈ V → ∪ dom 𝐹 ∈ V) | |
5 | 2, 3, 4 | 3syl 18 | . 2 ⊢ (𝜑 → ∪ dom 𝐹 ∈ V) |
6 | 1, 5 | syl5eqel 2705 | 1 ⊢ (𝜑 → 𝑋 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∪ cuni 4436 dom cdm 5114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-cnv 5122 df-dm 5124 df-rn 5125 |
This theorem is referenced by: omessle 40712 caragensplit 40714 omeunile 40719 caragenuncl 40727 omeunle 40730 omeiunlempt 40734 carageniuncllem2 40736 caragencmpl 40749 |
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