Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fobigcup Structured version   Visualization version   GIF version

Theorem fobigcup 33264
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 7461 . . . 4 (𝑥 ∈ V → 𝑥 ∈ V)
21rgen 3153 . . 3 𝑥 ∈ V 𝑥 ∈ V
3 dfbigcup2 33263 . . . 4 Bigcup = (𝑥 ∈ V ↦ 𝑥)
43mptfng 6486 . . 3 (∀𝑥 ∈ V 𝑥 ∈ V ↔ Bigcup Fn V)
52, 4mpbi 231 . 2 Bigcup Fn V
63rnmpt 5826 . . 3 ran Bigcup = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝑥}
7 vex 3503 . . . . 5 𝑦 ∈ V
8 snex 5328 . . . . . 6 {𝑦} ∈ V
97unisn 4853 . . . . . . 7 {𝑦} = 𝑦
109eqcomi 2835 . . . . . 6 𝑦 = {𝑦}
11 unieq 4845 . . . . . . 7 (𝑥 = {𝑦} → 𝑥 = {𝑦})
1211rspceeqv 3642 . . . . . 6 (({𝑦} ∈ V ∧ 𝑦 = {𝑦}) → ∃𝑥 ∈ V 𝑦 = 𝑥)
138, 10, 12mp2an 688 . . . . 5 𝑥 ∈ V 𝑦 = 𝑥
147, 132th 265 . . . 4 (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = 𝑥)
1514abbi2i 2958 . . 3 V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝑥}
166, 15eqtr4i 2852 . 2 ran Bigcup = V
17 df-fo 6360 . 2 ( Bigcup :V–onto→V ↔ ( Bigcup Fn V ∧ ran Bigcup = V))
185, 16, 17mpbir2an 707 1 Bigcup :V–onto→V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  wcel 2107  {cab 2804  wral 3143  wrex 3144  Vcvv 3500  {csn 4564   cuni 4837  ran crn 5555   Fn wfn 6349  ontowfo 6352   Bigcup cbigcup 33198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-symdif 4223  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-eprel 5464  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fo 6360  df-fv 6362  df-1st 7685  df-2nd 7686  df-txp 33218  df-bigcup 33222
This theorem is referenced by:  fnbigcup  33265
  Copyright terms: Public domain W3C validator