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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvbigcup | Structured version Visualization version GIF version |
Description: For sets, Bigcup yields union. (Contributed by Scott Fenton, 11-Apr-2012.) |
Ref | Expression |
---|---|
fvbigcup.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
fvbigcup | ⊢ ( Bigcup ‘𝐴) = ∪ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝐴 | |
2 | fvbigcup.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
3 | 2 | uniex 6851 | . . . 4 ⊢ ∪ 𝐴 ∈ V |
4 | 3 | brbigcup 31175 | . . 3 ⊢ (𝐴 Bigcup ∪ 𝐴 ↔ ∪ 𝐴 = ∪ 𝐴) |
5 | 1, 4 | mpbir 220 | . 2 ⊢ 𝐴 Bigcup ∪ 𝐴 |
6 | fnbigcup 31178 | . . 3 ⊢ Bigcup Fn V | |
7 | fnbrfvb 6146 | . . 3 ⊢ (( Bigcup Fn V ∧ 𝐴 ∈ V) → (( Bigcup ‘𝐴) = ∪ 𝐴 ↔ 𝐴 Bigcup ∪ 𝐴)) | |
8 | 6, 2, 7 | mp2an 704 | . 2 ⊢ (( Bigcup ‘𝐴) = ∪ 𝐴 ↔ 𝐴 Bigcup ∪ 𝐴) |
9 | 5, 8 | mpbir 220 | 1 ⊢ ( Bigcup ‘𝐴) = ∪ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cuni 4372 class class class wbr 4583 Fn wfn 5799 ‘cfv 5804 Bigcup cbigcup 31110 |
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-pow 4769 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-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-symdif 3806 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-eprel 4949 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-1st 7059 df-2nd 7060 df-txp 31130 df-bigcup 31134 |
This theorem is referenced by: (None) |
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