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Theorem fo2nd 5785
 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 2560 . . . . . 6 𝑥 ∈ V
2 snexgOLD 3935 . . . . . 6 (𝑥 ∈ V → {𝑥} ∈ V)
31, 2ax-mp 7 . . . . 5 {𝑥} ∈ V
43rnex 4599 . . . 4 ran {𝑥} ∈ V
54uniex 4174 . . 3 ran {𝑥} ∈ V
6 df-2nd 5768 . . 3 2nd = (𝑥 ∈ V ↦ ran {𝑥})
75, 6fnmpti 5027 . 2 2nd Fn V
86rnmpt 4582 . . 3 ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ran {𝑥}}
9 vex 2560 . . . . 5 𝑦 ∈ V
109, 9opex 3966 . . . . . 6 𝑦, 𝑦⟩ ∈ V
119, 9op2nda 4805 . . . . . . 7 ran {⟨𝑦, 𝑦⟩} = 𝑦
1211eqcomi 2044 . . . . . 6 𝑦 = ran {⟨𝑦, 𝑦⟩}
13 sneq 3386 . . . . . . . . . 10 (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
1413rneqd 4563 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
1514unieqd 3591 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
1615eqeq2d 2051 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑦⟩ → (𝑦 = ran {𝑥} ↔ 𝑦 = ran {⟨𝑦, 𝑦⟩}))
1716rspcev 2656 . . . . . 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 2152 . . 3 V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ran {𝑥}}
218, 20eqtr4i 2063 . 2 ran 2nd = V
22 df-fo 4908 . 2 (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V))
237, 21, 22mpbir2an 849 1 2nd :V–onto→V
 Colors of variables: wff set class Syntax hints:   = wceq 1243   ∈ wcel 1393  {cab 2026  ∃wrex 2307  Vcvv 2557  {csn 3375  ⟨cop 3378  ∪ cuni 3580  ran crn 4346   Fn wfn 4897  –onto→wfo 4900  2nd c2nd 5766 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-fo 4908  df-2nd 5768 This theorem is referenced by:  2ndcof  5791  2ndexg  5795  df2nd2  5841  2ndconst  5843
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