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Theorem fo2nd 7149
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 snex 4879 . . . . 5 {𝑥} ∈ V
21rnex 7062 . . . 4 ran {𝑥} ∈ V
32uniex 6918 . . 3 ran {𝑥} ∈ V
4 df-2nd 7129 . . 3 2nd = (𝑥 ∈ V ↦ ran {𝑥})
53, 4fnmpti 5989 . 2 2nd Fn V
64rnmpt 5341 . . 3 ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ran {𝑥}}
7 vex 3193 . . . . 5 𝑦 ∈ V
8 opex 4903 . . . . . 6 𝑦, 𝑦⟩ ∈ V
97, 7op2nda 5589 . . . . . . 7 ran {⟨𝑦, 𝑦⟩} = 𝑦
109eqcomi 2630 . . . . . 6 𝑦 = ran {⟨𝑦, 𝑦⟩}
11 sneq 4165 . . . . . . . . . 10 (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
1211rneqd 5323 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
1312unieqd 4419 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
1413eqeq2d 2631 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑦⟩ → (𝑦 = ran {𝑥} ↔ 𝑦 = ran {⟨𝑦, 𝑦⟩}))
1514rspcev 3299 . . . . . 6 ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ran {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ran {𝑥})
168, 10, 15mp2an 707 . . . . 5 𝑥 ∈ V 𝑦 = ran {𝑥}
177, 162th 254 . . . 4 (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ran {𝑥})
1817abbi2i 2735 . . 3 V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ran {𝑥}}
196, 18eqtr4i 2646 . 2 ran 2nd = V
20 df-fo 5863 . 2 (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V))
215, 19, 20mpbir2an 954 1 2nd :V–onto→V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  {cab 2607  wrex 2909  Vcvv 3190  {csn 4155  cop 4161   cuni 4409  ran crn 5085   Fn wfn 5852  ontowfo 5855  2nd c2nd 7127
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 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
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-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-fun 5859  df-fn 5860  df-fo 5863  df-2nd 7129
This theorem is referenced by:  2ndcof  7157  df2nd2  7224  2ndconst  7226  iunfo  9321  cdaf  16640  2ndf1  16775  2ndf2  16776  2ndfcl  16778  gsum2dlem2  18310  upxp  21366  uptx  21368  cnmpt2nd  21412  uniiccdif  23286  xppreima  29332  xppreima2  29333  2ndpreima  29369  gsummpt2d  29608  cnre2csqima  29781  br2ndeq  31428  filnetlem4  32071
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