ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fo2nd GIF version

Theorem fo2nd 5816
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 2605 . . . . . 6 𝑥 ∈ V
21snex 3965 . . . . 5 {𝑥} ∈ V
32rnex 4627 . . . 4 ran {𝑥} ∈ V
43uniex 4200 . . 3 ran {𝑥} ∈ V
5 df-2nd 5799 . . 3 2nd = (𝑥 ∈ V ↦ ran {𝑥})
64, 5fnmpti 5058 . 2 2nd Fn V
75rnmpt 4610 . . 3 ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ran {𝑥}}
8 vex 2605 . . . . 5 𝑦 ∈ V
98, 8opex 3992 . . . . . 6 𝑦, 𝑦⟩ ∈ V
108, 8op2nda 4835 . . . . . . 7 ran {⟨𝑦, 𝑦⟩} = 𝑦
1110eqcomi 2086 . . . . . 6 𝑦 = ran {⟨𝑦, 𝑦⟩}
12 sneq 3417 . . . . . . . . . 10 (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
1312rneqd 4591 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
1413unieqd 3620 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
1514eqeq2d 2093 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑦⟩ → (𝑦 = ran {𝑥} ↔ 𝑦 = ran {⟨𝑦, 𝑦⟩}))
1615rspcev 2702 . . . . . 6 ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ran {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ran {𝑥})
179, 11, 16mp2an 417 . . . . 5 𝑥 ∈ V 𝑦 = ran {𝑥}
188, 172th 172 . . . 4 (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ran {𝑥})
1918abbi2i 2194 . . 3 V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ran {𝑥}}
207, 19eqtr4i 2105 . 2 ran 2nd = V
21 df-fo 4938 . 2 (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V))
226, 20, 21mpbir2an 884 1 2nd :V–onto→V
Colors of variables: wff set class
Syntax hints:   = wceq 1285  wcel 1434  {cab 2068  wrex 2350  Vcvv 2602  {csn 3406  cop 3409   cuni 3609  ran crn 4372   Fn wfn 4927  ontowfo 4930  2nd c2nd 5797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-fun 4934  df-fn 4935  df-fo 4938  df-2nd 5799
This theorem is referenced by:  2ndcof  5822  2ndexg  5826  df2nd2  5872  2ndconst  5874
  Copyright terms: Public domain W3C validator