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Theorem fo2nd 7387
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 5064 . . . . 5 {𝑥} ∈ V
21rnex 7298 . . . 4 ran {𝑥} ∈ V
32uniex 7151 . . 3 ran {𝑥} ∈ V
4 df-2nd 7367 . . 3 2nd = (𝑥 ∈ V ↦ ran {𝑥})
53, 4fnmpti 6200 . 2 2nd Fn V
64rnmpt 5540 . . 3 ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ran {𝑥}}
7 vex 3353 . . . . 5 𝑦 ∈ V
8 opex 5088 . . . . . 6 𝑦, 𝑦⟩ ∈ V
97, 7op2nda 5806 . . . . . . 7 ran {⟨𝑦, 𝑦⟩} = 𝑦
109eqcomi 2774 . . . . . 6 𝑦 = ran {⟨𝑦, 𝑦⟩}
11 sneq 4344 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
1211rneqd 5521 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
1312unieqd 4604 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
1413rspceeqv 3479 . . . . . 6 ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ran {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ran {𝑥})
158, 10, 14mp2an 683 . . . . 5 𝑥 ∈ V 𝑦 = ran {𝑥}
167, 152th 255 . . . 4 (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ran {𝑥})
1716abbi2i 2881 . . 3 V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ran {𝑥}}
186, 17eqtr4i 2790 . 2 ran 2nd = V
19 df-fo 6074 . 2 (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V))
205, 18, 19mpbir2an 702 1 2nd :V–onto→V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  wcel 2155  {cab 2751  wrex 3056  Vcvv 3350  {csn 4334  cop 4340   cuni 4594  ran crn 5278   Fn wfn 6063  ontowfo 6066  2nd c2nd 7365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-fun 6070  df-fn 6071  df-fo 6074  df-2nd 7367
This theorem is referenced by:  br2ndeqg  7389  2ndcof  7397  df2nd2  7466  2ndconst  7468  iunfo  9614  cdaf  16967  2ndf1  17103  2ndf2  17104  2ndfcl  17106  gsum2dlem2  18636  upxp  21706  uptx  21708  cnmpt2nd  21752  uniiccdif  23636  xppreima  29834  xppreima2  29835  2ndpreima  29869  gsummpt2d  30163  cnre2csqima  30339  filnetlem4  32751
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