![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
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 2736 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝐴 | |
2 | fvbigcup.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
3 | 2 | uniex 7675 | . . . 4 ⊢ ∪ 𝐴 ∈ V |
4 | 3 | brbigcup 34472 | . . 3 ⊢ (𝐴 Bigcup ∪ 𝐴 ↔ ∪ 𝐴 = ∪ 𝐴) |
5 | 1, 4 | mpbir 230 | . 2 ⊢ 𝐴 Bigcup ∪ 𝐴 |
6 | fnbigcup 34475 | . . 3 ⊢ Bigcup Fn V | |
7 | fnbrfvb 6893 | . . 3 ⊢ (( Bigcup Fn V ∧ 𝐴 ∈ V) → (( Bigcup ‘𝐴) = ∪ 𝐴 ↔ 𝐴 Bigcup ∪ 𝐴)) | |
8 | 6, 2, 7 | mp2an 690 | . 2 ⊢ (( Bigcup ‘𝐴) = ∪ 𝐴 ↔ 𝐴 Bigcup ∪ 𝐴) |
9 | 5, 8 | mpbir 230 | 1 ⊢ ( Bigcup ‘𝐴) = ∪ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∈ wcel 2106 Vcvv 3444 ∪ cuni 4864 class class class wbr 5104 Fn wfn 6489 ‘cfv 6494 Bigcup cbigcup 34408 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-symdif 4201 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-eprel 5536 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-fo 6500 df-fv 6502 df-1st 7918 df-2nd 7919 df-txp 34428 df-bigcup 34432 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |