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Theorem fo2nd 5704
 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 vex 2534 . . . . . 6 x V
2 snexgOLD 3905 . . . . . 6 (x V → {x} V)
31, 2ax-mp 7 . . . . 5 {x} V
43rnex 4522 . . . 4 ran {x} V
54uniex 4120 . . 3 ran {x} V
6 df-2nd 5687 . . 3 2nd = (x V ↦ ran {x})
75, 6fnmpti 4949 . 2 2nd Fn V
86rnmpt 4505 . . 3 ran 2nd = {yx V y = ran {x}}
9 vex 2534 . . . . 5 y V
109, 9opex 3936 . . . . . 6 y, y V
119, 9op2nda 4728 . . . . . . 7 ran {⟨y, y⟩} = y
1211eqcomi 2022 . . . . . 6 y = ran {⟨y, y⟩}
13 sneq 3357 . . . . . . . . . 10 (x = ⟨y, y⟩ → {x} = {⟨y, y⟩})
1413rneqd 4486 . . . . . . . . 9 (x = ⟨y, y⟩ → ran {x} = ran {⟨y, y⟩})
1514unieqd 3561 . . . . . . . 8 (x = ⟨y, y⟩ → ran {x} = ran {⟨y, y⟩})
1615eqeq2d 2029 . . . . . . 7 (x = ⟨y, y⟩ → (y = ran {x} ↔ y = ran {⟨y, y⟩}))
1716rspcev 2629 . . . . . 6 ((⟨y, y V y = ran {⟨y, y⟩}) → x V y = ran {x})
1810, 12, 17mp2an 404 . . . . 5 x V y = ran {x}
199, 182th 163 . . . 4 (y V ↔ x V y = ran {x})
2019abbi2i 2130 . . 3 V = {yx V y = ran {x}}
218, 20eqtr4i 2041 . 2 ran 2nd = V
22 df-fo 4831 . 2 (2nd :V–onto→V ↔ (2nd Fn V ran 2nd = V))
237, 21, 22mpbir2an 835 1 2nd :V–onto→V
 Colors of variables: wff set class Syntax hints:   = wceq 1226   ∈ wcel 1370  {cab 2004  ∃wrex 2281  Vcvv 2531  {csn 3346  ⟨cop 3349  ∪ cuni 3550  ran crn 4269   Fn wfn 4820  –onto→wfo 4823  2nd c2nd 5685 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-un 4116 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-fun 4827  df-fn 4828  df-fo 4831  df-2nd 5687 This theorem is referenced by:  2ndcof  5710  2ndexg  5714  df2nd2  5760  2ndconst  5762
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