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Theorem fo2nd 7450
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 snex 5130 . . . . 5 {𝑥} ∈ V
21rnex 7363 . . . 4 ran {𝑥} ∈ V
32uniex 7214 . . 3 ran {𝑥} ∈ V
4 df-2nd 7430 . . 3 2nd = (𝑥 ∈ V ↦ ran {𝑥})
53, 4fnmpti 6256 . 2 2nd Fn V
64rnmpt 5605 . . 3 ran 2nd = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ran {𝑥}}
7 vex 3418 . . . . 5 𝑦 ∈ V
8 opex 5154 . . . . . 6 𝑦, 𝑦⟩ ∈ V
97, 7op2nda 5863 . . . . . . 7 ran {⟨𝑦, 𝑦⟩} = 𝑦
109eqcomi 2835 . . . . . 6 𝑦 = ran {⟨𝑦, 𝑦⟩}
11 sneq 4408 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
1211rneqd 5586 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
1312unieqd 4669 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑦⟩ → ran {𝑥} = ran {⟨𝑦, 𝑦⟩})
1413rspceeqv 3545 . . . . . 6 ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ran {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ran {𝑥})
158, 10, 14mp2an 685 . . . . 5 𝑥 ∈ V 𝑦 = ran {𝑥}
167, 152th 256 . . . 4 (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ran {𝑥})
1716abbi2i 2944 . . 3 V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ran {𝑥}}
186, 17eqtr4i 2853 . 2 ran 2nd = V
19 df-fo 6130 . 2 (2nd :V–onto→V ↔ (2nd Fn V ∧ ran 2nd = V))
205, 18, 19mpbir2an 704 1 2nd :V–onto→V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1658  wcel 2166  {cab 2812  wrex 3119  Vcvv 3415  {csn 4398  cop 4404   cuni 4659  ran crn 5344   Fn wfn 6119  ontowfo 6122  2nd c2nd 7428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128  ax-un 7210
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-fun 6126  df-fn 6127  df-fo 6130  df-2nd 7430
This theorem is referenced by:  br2ndeqg  7452  2ndcof  7460  df2nd2  7529  2ndconst  7531  iunfo  9677  cdaf  17053  2ndf1  17189  2ndf2  17190  2ndfcl  17192  gsum2dlem2  18724  upxp  21798  uptx  21800  cnmpt2nd  21844  uniiccdif  23745  xppreima  29999  xppreima2  30000  2ndpreima  30034  gsummpt2d  30327  cnre2csqima  30503  filnetlem4  32915
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