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Theorem fobigcup 31646
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 6908 . . . 4 (𝑥 ∈ V → 𝑥 ∈ V)
21rgen 2917 . . 3 𝑥 ∈ V 𝑥 ∈ V
3 dfbigcup2 31645 . . . 4 Bigcup = (𝑥 ∈ V ↦ 𝑥)
43mptfng 5976 . . 3 (∀𝑥 ∈ V 𝑥 ∈ V ↔ Bigcup Fn V)
52, 4mpbi 220 . 2 Bigcup Fn V
63rnmpt 5331 . . 3 ran Bigcup = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝑥}
7 vex 3189 . . . . 5 𝑦 ∈ V
8 snex 4869 . . . . . 6 {𝑦} ∈ V
97unisn 4417 . . . . . . 7 {𝑦} = 𝑦
109eqcomi 2630 . . . . . 6 𝑦 = {𝑦}
11 unieq 4410 . . . . . . . 8 (𝑥 = {𝑦} → 𝑥 = {𝑦})
1211eqeq2d 2631 . . . . . . 7 (𝑥 = {𝑦} → (𝑦 = 𝑥𝑦 = {𝑦}))
1312rspcev 3295 . . . . . 6 (({𝑦} ∈ V ∧ 𝑦 = {𝑦}) → ∃𝑥 ∈ V 𝑦 = 𝑥)
148, 10, 13mp2an 707 . . . . 5 𝑥 ∈ V 𝑦 = 𝑥
157, 142th 254 . . . 4 (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = 𝑥)
1615abbi2i 2735 . . 3 V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝑥}
176, 16eqtr4i 2646 . 2 ran Bigcup = V
18 df-fo 5853 . 2 ( Bigcup :V–onto→V ↔ ( Bigcup Fn V ∧ ran Bigcup = V))
195, 17, 18mpbir2an 954 1 Bigcup :V–onto→V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  {cab 2607  wral 2907  wrex 2908  Vcvv 3186  {csn 4148   cuni 4402  ran crn 5075   Fn wfn 5842  ontowfo 5845   Bigcup cbigcup 31579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-symdif 3822  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-eprel 4985  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fo 5853  df-fv 5855  df-1st 7113  df-2nd 7114  df-txp 31599  df-bigcup 31603
This theorem is referenced by:  fnbigcup  31647
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