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Theorem fo2nd 5810
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 2575 . . . . . 6 𝑥 ∈ V
2 snexgOLD 3960 . . . . . 6 (𝑥 ∈ V → {𝑥} ∈ V)
31, 2ax-mp 7 . . . . 5 {𝑥} ∈ V
43rnex 4624 . . . 4 ran {𝑥} ∈ V
54uniex 4199 . . 3 ran {𝑥} ∈ V
6 df-2nd 5793 . . 3 2nd = (𝑥 ∈ V ↦ ran {𝑥})
75, 6fnmpti 5052 . 2 2nd Fn V
86rnmpt 4607 . . 3 ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ran {𝑥}}
9 vex 2575 . . . . 5 𝑦 ∈ V
109, 9opex 3991 . . . . . 6 𝑦, 𝑦⟩ ∈ V
119, 9op2nda 4830 . . . . . . 7 ran {⟨𝑦, 𝑦⟩} = 𝑦
1211eqcomi 2058 . . . . . 6 𝑦 = ran {⟨𝑦, 𝑦⟩}
13 sneq 3411 . . . . . . . . . 10 (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
1413rneqd 4588 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
1514unieqd 3616 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
1615eqeq2d 2065 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑦⟩ → (𝑦 = ran {𝑥} ↔ 𝑦 = ran {⟨𝑦, 𝑦⟩}))
1716rspcev 2671 . . . . . 6 ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ran {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ran {𝑥})
1810, 12, 17mp2an 410 . . . . 5 𝑥 ∈ V 𝑦 = ran {𝑥}
199, 182th 167 . . . 4 (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ran {𝑥})
2019abbi2i 2166 . . 3 V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ran {𝑥}}
218, 20eqtr4i 2077 . 2 ran 2nd = V
22 df-fo 4933 . 2 (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V))
237, 21, 22mpbir2an 858 1 2nd :V–onto→V
Colors of variables: wff set class
Syntax hints:   = wceq 1257  wcel 1407  {cab 2040  wrex 2322  Vcvv 2572  {csn 3400  cop 3403   cuni 3605  ran crn 4371   Fn wfn 4922  ontowfo 4925  2nd c2nd 5791
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969  ax-un 4195
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-br 3790  df-opab 3844  df-mpt 3845  df-id 4055  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-fun 4929  df-fn 4930  df-fo 4933  df-2nd 5793
This theorem is referenced by:  2ndcof  5816  2ndexg  5820  df2nd2  5866  2ndconst  5868
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