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Theorem fo2nd 7704
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 5327 . . . . 5 {𝑥} ∈ V
21rnex 7608 . . . 4 ran {𝑥} ∈ V
32uniex 7458 . . 3 ran {𝑥} ∈ V
4 df-2nd 7684 . . 3 2nd = (𝑥 ∈ V ↦ ran {𝑥})
53, 4fnmpti 6487 . 2 2nd Fn V
64rnmpt 5825 . . 3 ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ran {𝑥}}
7 vex 3502 . . . . 5 𝑦 ∈ V
8 opex 5352 . . . . . 6 𝑦, 𝑦⟩ ∈ V
97, 7op2nda 6082 . . . . . . 7 ran {⟨𝑦, 𝑦⟩} = 𝑦
109eqcomi 2833 . . . . . 6 𝑦 = ran {⟨𝑦, 𝑦⟩}
11 sneq 4573 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
1211rneqd 5806 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
1312unieqd 4846 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
1413rspceeqv 3641 . . . . . 6 ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ran {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ran {𝑥})
158, 10, 14mp2an 688 . . . . 5 𝑥 ∈ V 𝑦 = ran {𝑥}
167, 152th 265 . . . 4 (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ran {𝑥})
1716abbi2i 2957 . . 3 V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ran {𝑥}}
186, 17eqtr4i 2851 . 2 ran 2nd = V
19 df-fo 6357 . 2 (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V))
205, 18, 19mpbir2an 707 1 2nd :V–onto→V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  wcel 2106  {cab 2802  wrex 3143  Vcvv 3499  {csn 4563  cop 4569   cuni 4836  ran crn 5554   Fn wfn 6346  ontowfo 6349  2nd c2nd 7682
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 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pr 5325  ax-un 7454
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 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-fun 6353  df-fn 6354  df-fo 6357  df-2nd 7684
This theorem is referenced by:  br2ndeqg  7706  2ndcof  7714  df2nd2  7788  2ndconst  7790  iunfo  9953  cdaf  17302  2ndf1  17437  2ndf2  17438  2ndfcl  17440  gsum2dlem2  19013  upxp  22147  uptx  22149  cnmpt2nd  22193  uniiccdif  24094  xppreima  30310  xppreima2  30311  2ndpreima  30357  fsuppcurry1  30375  gsummpt2d  30602  cnre2csqima  31041  filnetlem4  33614
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