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Theorem fvbigcup 32543
Description: For sets, Bigcup yields union. (Contributed by Scott Fenton, 11-Apr-2012.)
Hypothesis
Ref Expression
fvbigcup.1 𝐴 ∈ V
Assertion
Ref Expression
fvbigcup ( Bigcup 𝐴) = 𝐴

Proof of Theorem fvbigcup
StepHypRef Expression
1 eqid 2825 . . 3 𝐴 = 𝐴
2 fvbigcup.1 . . . . 5 𝐴 ∈ V
32uniex 7218 . . . 4 𝐴 ∈ V
43brbigcup 32539 . . 3 (𝐴 Bigcup 𝐴 𝐴 = 𝐴)
51, 4mpbir 223 . 2 𝐴 Bigcup 𝐴
6 fnbigcup 32542 . . 3 Bigcup Fn V
7 fnbrfvb 6486 . . 3 (( Bigcup Fn V ∧ 𝐴 ∈ V) → (( Bigcup 𝐴) = 𝐴𝐴 Bigcup 𝐴))
86, 2, 7mp2an 683 . 2 (( Bigcup 𝐴) = 𝐴𝐴 Bigcup 𝐴)
95, 8mpbir 223 1 ( Bigcup 𝐴) = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1656  wcel 2164  Vcvv 3414   cuni 4660   class class class wbr 4875   Fn wfn 6122  cfv 6127   Bigcup cbigcup 32475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-symdif 4072  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-eprel 5257  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fo 6133  df-fv 6135  df-1st 7433  df-2nd 7434  df-txp 32495  df-bigcup 32499
This theorem is referenced by: (None)
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