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Theorem fo2nd 8035
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 5434 . . . . 5 {𝑥} ∈ V
21rnex 7932 . . . 4 ran {𝑥} ∈ V
32uniex 7761 . . 3 ran {𝑥} ∈ V
4 df-2nd 8015 . . 3 2nd = (𝑥 ∈ V ↦ ran {𝑥})
53, 4fnmpti 6711 . 2 2nd Fn V
64rnmpt 5968 . . 3 ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ran {𝑥}}
7 vex 3484 . . . . 5 𝑦 ∈ V
8 opex 5469 . . . . . 6 𝑦, 𝑦⟩ ∈ V
97, 7op2nda 6248 . . . . . . 7 ran {⟨𝑦, 𝑦⟩} = 𝑦
109eqcomi 2746 . . . . . 6 𝑦 = ran {⟨𝑦, 𝑦⟩}
11 sneq 4636 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
1211rneqd 5949 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
1312unieqd 4920 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
1413rspceeqv 3645 . . . . . 6 ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ran {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ran {𝑥})
158, 10, 14mp2an 692 . . . . 5 𝑥 ∈ V 𝑦 = ran {𝑥}
167, 152th 264 . . . 4 (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ran {𝑥})
1716eqabi 2877 . . 3 V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ran {𝑥}}
186, 17eqtr4i 2768 . 2 ran 2nd = V
19 df-fo 6567 . 2 (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V))
205, 18, 19mpbir2an 711 1 2nd :V–onto→V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  Vcvv 3480  {csn 4626  cop 4632   cuni 4907  ran crn 5686   Fn wfn 6556  ontowfo 6559  2nd c2nd 8013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-fo 6567  df-2nd 8015
This theorem is referenced by:  br2ndeqg  8037  2ndcof  8045  df2nd2  8124  2ndconst  8126  opco2  8149  iunfo  10579  cdaf  18095  2ndf1  18240  2ndf2  18241  2ndfcl  18243  gsum2dlem2  19989  upxp  23631  uptx  23633  cnmpt2nd  23677  uniiccdif  25613  precsexlem10  28240  precsexlem11  28241  xppreima  32655  2ndimaxp  32656  2ndresdju  32659  xppreima2  32661  2ndpreima  32717  fsuppcurry1  32736  gsummpt2d  33052  gsumpart  33060  cnre2csqima  33910  filnetlem4  36382
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