MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fo2nd Structured version   Visualization version   GIF version

Theorem fo2nd 8026
Description: The 2nd function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
fo2nd 2nd :V–onto→V

Proof of Theorem fo2nd
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vsnex 5437 . . . . 5 {𝑥} ∈ V
21rnex 7925 . . . 4 ran {𝑥} ∈ V
32uniex 7754 . . 3 ran {𝑥} ∈ V
4 df-2nd 8006 . . 3 2nd = (𝑥 ∈ V ↦ ran {𝑥})
53, 4fnmpti 6706 . 2 2nd Fn V
64rnmpt 5963 . . 3 ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ran {𝑥}}
7 vex 3466 . . . . 5 𝑦 ∈ V
8 opex 5472 . . . . . 6 𝑦, 𝑦⟩ ∈ V
97, 7op2nda 6241 . . . . . . 7 ran {⟨𝑦, 𝑦⟩} = 𝑦
109eqcomi 2735 . . . . . 6 𝑦 = ran {⟨𝑦, 𝑦⟩}
11 sneq 4643 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
1211rneqd 5946 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
1312unieqd 4928 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
1413rspceeqv 3630 . . . . . 6 ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ran {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ran {𝑥})
158, 10, 14mp2an 690 . . . . 5 𝑥 ∈ V 𝑦 = ran {𝑥}
167, 152th 263 . . . 4 (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ran {𝑥})
1716eqabi 2862 . . 3 V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ran {𝑥}}
186, 17eqtr4i 2757 . 2 ran 2nd = V
19 df-fo 6562 . 2 (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V))
205, 18, 19mpbir2an 709 1 2nd :V–onto→V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wcel 2099  {cab 2703  wrex 3060  Vcvv 3462  {csn 4633  cop 4639   cuni 4915  ran crn 5685   Fn wfn 6551  ontowfo 6554  2nd c2nd 8004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5306  ax-nul 5313  ax-pr 5435  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-br 5156  df-opab 5218  df-mpt 5239  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-fun 6558  df-fn 6559  df-fo 6562  df-2nd 8006
This theorem is referenced by:  br2ndeqg  8028  2ndcof  8036  df2nd2  8115  2ndconst  8117  opco2  8140  iunfo  10584  cdaf  18074  2ndf1  18221  2ndf2  18222  2ndfcl  18224  gsum2dlem2  19971  upxp  23621  uptx  23623  cnmpt2nd  23667  uniiccdif  25601  precsexlem10  28218  precsexlem11  28219  xppreima  32565  2ndimaxp  32566  2ndresdju  32568  xppreima2  32570  2ndpreima  32621  fsuppcurry1  32641  gsummpt2d  32919  gsumpart  32925  cnre2csqima  33728  filnetlem4  36095
  Copyright terms: Public domain W3C validator