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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 2793 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝐴 | |
2 | fvbigcup.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
3 | 2 | uniex 7314 | . . . 4 ⊢ ∪ 𝐴 ∈ V |
4 | 3 | brbigcup 32913 | . . 3 ⊢ (𝐴 Bigcup ∪ 𝐴 ↔ ∪ 𝐴 = ∪ 𝐴) |
5 | 1, 4 | mpbir 232 | . 2 ⊢ 𝐴 Bigcup ∪ 𝐴 |
6 | fnbigcup 32916 | . . 3 ⊢ Bigcup Fn V | |
7 | fnbrfvb 6578 | . . 3 ⊢ (( Bigcup Fn V ∧ 𝐴 ∈ V) → (( Bigcup ‘𝐴) = ∪ 𝐴 ↔ 𝐴 Bigcup ∪ 𝐴)) | |
8 | 6, 2, 7 | mp2an 688 | . 2 ⊢ (( Bigcup ‘𝐴) = ∪ 𝐴 ↔ 𝐴 Bigcup ∪ 𝐴) |
9 | 5, 8 | mpbir 232 | 1 ⊢ ( Bigcup ‘𝐴) = ∪ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1520 ∈ wcel 2079 Vcvv 3432 ∪ cuni 4739 class class class wbr 4956 Fn wfn 6212 ‘cfv 6217 Bigcup cbigcup 32849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-ral 3108 df-rex 3109 df-rab 3112 df-v 3434 df-sbc 3702 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-symdif 4134 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-op 4473 df-uni 4740 df-br 4957 df-opab 5019 df-mpt 5036 df-id 5340 df-eprel 5345 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-fo 6223 df-fv 6225 df-1st 7536 df-2nd 7537 df-txp 32869 df-bigcup 32873 |
This theorem is referenced by: (None) |
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