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Theorem fo2nd 5727
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 2554 . . . . . 6 x V
2 snexgOLD 3926 . . . . . 6 (x V → {x} V)
31, 2ax-mp 7 . . . . 5 {x} V
43rnex 4542 . . . 4 ran {x} V
54uniex 4140 . . 3 ran {x} V
6 df-2nd 5710 . . 3 2nd = (x V ↦ ran {x})
75, 6fnmpti 4970 . 2 2nd Fn V
86rnmpt 4525 . . 3 ran 2nd = {yx V y = ran {x}}
9 vex 2554 . . . . 5 y V
109, 9opex 3957 . . . . . 6 y, y V
119, 9op2nda 4748 . . . . . . 7 ran {⟨y, y⟩} = y
1211eqcomi 2041 . . . . . 6 y = ran {⟨y, y⟩}
13 sneq 3378 . . . . . . . . . 10 (x = ⟨y, y⟩ → {x} = {⟨y, y⟩})
1413rneqd 4506 . . . . . . . . 9 (x = ⟨y, y⟩ → ran {x} = ran {⟨y, y⟩})
1514unieqd 3582 . . . . . . . 8 (x = ⟨y, y⟩ → ran {x} = ran {⟨y, y⟩})
1615eqeq2d 2048 . . . . . . 7 (x = ⟨y, y⟩ → (y = ran {x} ↔ y = ran {⟨y, y⟩}))
1716rspcev 2650 . . . . . 6 ((⟨y, y V y = ran {⟨y, y⟩}) → x V y = ran {x})
1810, 12, 17mp2an 402 . . . . 5 x V y = ran {x}
199, 182th 163 . . . 4 (y V ↔ x V y = ran {x})
2019abbi2i 2149 . . 3 V = {yx V y = ran {x}}
218, 20eqtr4i 2060 . 2 ran 2nd = V
22 df-fo 4851 . 2 (2nd :V–onto→V ↔ (2nd Fn V ran 2nd = V))
237, 21, 22mpbir2an 848 1 2nd :V–onto→V
Colors of variables: wff set class
Syntax hints:   = wceq 1242   wcel 1390  {cab 2023  wrex 2301  Vcvv 2551  {csn 3367  cop 3370   cuni 3571  ran crn 4289   Fn wfn 4840  ontowfo 4843  2nd c2nd 5708
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-fo 4851  df-2nd 5710
This theorem is referenced by:  2ndcof  5733  2ndexg  5737  df2nd2  5783  2ndconst  5785
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