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Theorem fobigcup 34860
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 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uniexg 7726 . . . 4 (𝑥 ∈ V → 𝑥 ∈ V)
21rgen 3063 . . 3 𝑥 ∈ V 𝑥 ∈ V
3 dfbigcup2 34859 . . . 4 Bigcup = (𝑥 ∈ V ↦ 𝑥)
43mptfng 6686 . . 3 (∀𝑥 ∈ V 𝑥 ∈ V ↔ Bigcup Fn V)
52, 4mpbi 229 . 2 Bigcup Fn V
63rnmpt 5952 . . 3 ran Bigcup = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝑥}
7 vex 3478 . . . . 5 𝑦 ∈ V
8 vsnex 5428 . . . . . 6 {𝑦} ∈ V
9 unisnv 4930 . . . . . . 7 {𝑦} = 𝑦
109eqcomi 2741 . . . . . 6 𝑦 = {𝑦}
11 unieq 4918 . . . . . . 7 (𝑥 = {𝑦} → 𝑥 = {𝑦})
1211rspceeqv 3632 . . . . . 6 (({𝑦} ∈ V ∧ 𝑦 = {𝑦}) → ∃𝑥 ∈ V 𝑦 = 𝑥)
138, 10, 12mp2an 690 . . . . 5 𝑥 ∈ V 𝑦 = 𝑥
147, 132th 263 . . . 4 (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = 𝑥)
1514eqabi 2869 . . 3 V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝑥}
166, 15eqtr4i 2763 . 2 ran Bigcup = V
17 df-fo 6546 . 2 ( Bigcup :V–onto→V ↔ ( Bigcup Fn V ∧ ran Bigcup = V))
185, 16, 17mpbir2an 709 1 Bigcup :V–onto→V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wcel 2106  {cab 2709  wral 3061  wrex 3070  Vcvv 3474  {csn 4627   cuni 4907  ran crn 5676   Fn wfn 6535  ontowfo 6538   Bigcup cbigcup 34794
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 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-symdif 4241  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-eprel 5579  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-1st 7971  df-2nd 7972  df-txp 34814  df-bigcup 34818
This theorem is referenced by:  fnbigcup  34861
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