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Theorem fo2nd 8033
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 vsnex 5439 . . . . 5 {𝑥} ∈ V
21rnex 7932 . . . 4 ran {𝑥} ∈ V
32uniex 7759 . . 3 ran {𝑥} ∈ V
4 df-2nd 8013 . . 3 2nd = (𝑥 ∈ V ↦ ran {𝑥})
53, 4fnmpti 6711 . 2 2nd Fn V
64rnmpt 5970 . . 3 ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ran {𝑥}}
7 vex 3481 . . . . 5 𝑦 ∈ V
8 opex 5474 . . . . . 6 𝑦, 𝑦⟩ ∈ V
97, 7op2nda 6249 . . . . . . 7 ran {⟨𝑦, 𝑦⟩} = 𝑦
109eqcomi 2743 . . . . . 6 𝑦 = ran {⟨𝑦, 𝑦⟩}
11 sneq 4640 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
1211rneqd 5951 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
1312unieqd 4924 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
1413rspceeqv 3644 . . . . . 6 ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ran {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ran {𝑥})
158, 10, 14mp2an 692 . . . . 5 𝑥 ∈ V 𝑦 = ran {𝑥}
167, 152th 264 . . . 4 (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ran {𝑥})
1716eqabi 2874 . . 3 V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ran {𝑥}}
186, 17eqtr4i 2765 . 2 ran 2nd = V
19 df-fo 6568 . 2 (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V))
205, 18, 19mpbir2an 711 1 2nd :V–onto→V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1536  wcel 2105  {cab 2711  wrex 3067  Vcvv 3477  {csn 4630  cop 4636   cuni 4911  ran crn 5689   Fn wfn 6557  ontowfo 6560  2nd c2nd 8011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-fun 6564  df-fn 6565  df-fo 6568  df-2nd 8013
This theorem is referenced by:  br2ndeqg  8035  2ndcof  8043  df2nd2  8122  2ndconst  8124  opco2  8147  iunfo  10576  cdaf  18103  2ndf1  18250  2ndf2  18251  2ndfcl  18253  gsum2dlem2  20003  upxp  23646  uptx  23648  cnmpt2nd  23692  uniiccdif  25626  precsexlem10  28254  precsexlem11  28255  xppreima  32661  2ndimaxp  32662  2ndresdju  32665  xppreima2  32667  2ndpreima  32722  fsuppcurry1  32742  gsummpt2d  33034  gsumpart  33042  cnre2csqima  33871  filnetlem4  36363
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