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Theorem fobigcup 32284
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 7108 . . . 4 (𝑥 ∈ V → 𝑥 ∈ V)
21rgen 3048 . . 3 𝑥 ∈ V 𝑥 ∈ V
3 dfbigcup2 32283 . . . 4 Bigcup = (𝑥 ∈ V ↦ 𝑥)
43mptfng 6168 . . 3 (∀𝑥 ∈ V 𝑥 ∈ V ↔ Bigcup Fn V)
52, 4mpbi 220 . 2 Bigcup Fn V
63rnmpt 5514 . . 3 ran Bigcup = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝑥}
7 vex 3331 . . . . 5 𝑦 ∈ V
8 snex 5045 . . . . . 6 {𝑦} ∈ V
97unisn 4591 . . . . . . 7 {𝑦} = 𝑦
109eqcomi 2757 . . . . . 6 𝑦 = {𝑦}
11 unieq 4584 . . . . . . . 8 (𝑥 = {𝑦} → 𝑥 = {𝑦})
1211eqeq2d 2758 . . . . . . 7 (𝑥 = {𝑦} → (𝑦 = 𝑥𝑦 = {𝑦}))
1312rspcev 3437 . . . . . 6 (({𝑦} ∈ V ∧ 𝑦 = {𝑦}) → ∃𝑥 ∈ V 𝑦 = 𝑥)
148, 10, 13mp2an 710 . . . . 5 𝑥 ∈ V 𝑦 = 𝑥
157, 142th 254 . . . 4 (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = 𝑥)
1615abbi2i 2864 . . 3 V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝑥}
176, 16eqtr4i 2773 . 2 ran Bigcup = V
18 df-fo 6043 . 2 ( Bigcup :V–onto→V ↔ ( Bigcup Fn V ∧ ran Bigcup = V))
195, 17, 18mpbir2an 993 1 Bigcup :V–onto→V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1620  wcel 2127  {cab 2734  wral 3038  wrex 3039  Vcvv 3328  {csn 4309   cuni 4576  ran crn 5255   Fn wfn 6032  ontowfo 6035   Bigcup cbigcup 32218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-symdif 3975  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-eprel 5167  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-fo 6043  df-fv 6045  df-1st 7321  df-2nd 7322  df-txp 32238  df-bigcup 32242
This theorem is referenced by:  fnbigcup  32285
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