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Theorem fobigcup 35177
Description: Bigcup maps the universe onto itself. (Contributed by Scott Fenton, 16-Apr-2012.)
Assertion
Ref Expression
fobigcup Bigcup :V–onto→V

Proof of Theorem fobigcup
Dummy variables š‘„ š‘¦ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uniexg 7734 . . . 4 (š‘„ ∈ V → ∪ š‘„ ∈ V)
21rgen 3062 . . 3 āˆ€š‘„ ∈ V ∪ š‘„ ∈ V
3 dfbigcup2 35176 . . . 4 Bigcup = (š‘„ ∈ V ↦ ∪ š‘„)
43mptfng 6689 . . 3 (āˆ€š‘„ ∈ V ∪ š‘„ ∈ V ↔ Bigcup Fn V)
52, 4mpbi 229 . 2 Bigcup Fn V
63rnmpt 5954 . . 3 ran Bigcup = {š‘¦ ∣ āˆƒš‘„ ∈ V š‘¦ = ∪ š‘„}
7 vex 3477 . . . . 5 š‘¦ ∈ V
8 vsnex 5429 . . . . . 6 {š‘¦} ∈ V
9 unisnv 4931 . . . . . . 7 ∪ {š‘¦} = š‘¦
109eqcomi 2740 . . . . . 6 š‘¦ = ∪ {š‘¦}
11 unieq 4919 . . . . . . 7 (š‘„ = {š‘¦} → ∪ š‘„ = ∪ {š‘¦})
1211rspceeqv 3633 . . . . . 6 (({š‘¦} ∈ V ∧ š‘¦ = ∪ {š‘¦}) → āˆƒš‘„ ∈ V š‘¦ = ∪ š‘„)
138, 10, 12mp2an 689 . . . . 5 āˆƒš‘„ ∈ V š‘¦ = ∪ š‘„
147, 132th 264 . . . 4 (š‘¦ ∈ V ↔ āˆƒš‘„ ∈ V š‘¦ = ∪ š‘„)
1514eqabi 2868 . . 3 V = {š‘¦ ∣ āˆƒš‘„ ∈ V š‘¦ = ∪ š‘„}
166, 15eqtr4i 2762 . 2 ran Bigcup = V
17 df-fo 6549 . 2 ( Bigcup :V–onto→V ↔ ( Bigcup Fn V ∧ ran Bigcup = V))
185, 16, 17mpbir2an 708 1 Bigcup :V–onto→V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540   ∈ wcel 2105  {cab 2708  āˆ€wral 3060  āˆƒwrex 3069  Vcvv 3473  {csn 4628  āˆŖ cuni 4908  ran crn 5677   Fn wfn 6538  ā€“onto→wfo 6541   Bigcup cbigcup 35111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-symdif 4242  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-eprel 5580  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 7979  df-2nd 7980  df-txp 35131  df-bigcup 35135
This theorem is referenced by:  fnbigcup  35178
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