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Theorem fo1st 6105
Description: The 1st function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
fo1st 1st :V–onto→V

Proof of Theorem fo1st
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2715 . . . . . 6 𝑥 ∈ V
21snex 4146 . . . . 5 {𝑥} ∈ V
32dmex 4852 . . . 4 dom {𝑥} ∈ V
43uniex 4397 . . 3 dom {𝑥} ∈ V
5 df-1st 6088 . . 3 1st = (𝑥 ∈ V ↦ dom {𝑥})
64, 5fnmpti 5298 . 2 1st Fn V
75rnmpt 4834 . . 3 ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = dom {𝑥}}
8 vex 2715 . . . . 5 𝑦 ∈ V
98, 8opex 4189 . . . . . 6 𝑦, 𝑦⟩ ∈ V
108, 8op1sta 5067 . . . . . . 7 dom {⟨𝑦, 𝑦⟩} = 𝑦
1110eqcomi 2161 . . . . . 6 𝑦 = dom {⟨𝑦, 𝑦⟩}
12 sneq 3571 . . . . . . . . . 10 (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
1312dmeqd 4788 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
1413unieqd 3783 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
1514eqeq2d 2169 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑦⟩ → (𝑦 = dom {𝑥} ↔ 𝑦 = dom {⟨𝑦, 𝑦⟩}))
1615rspcev 2816 . . . . . 6 ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = dom {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = dom {𝑥})
179, 11, 16mp2an 423 . . . . 5 𝑥 ∈ V 𝑦 = dom {𝑥}
188, 172th 173 . . . 4 (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = dom {𝑥})
1918abbi2i 2272 . . 3 V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = dom {𝑥}}
207, 19eqtr4i 2181 . 2 ran 1st = V
21 df-fo 5176 . 2 (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
226, 20, 21mpbir2an 927 1 1st :V–onto→V
Colors of variables: wff set class
Syntax hints:   = wceq 1335  wcel 2128  {cab 2143  wrex 2436  Vcvv 2712  {csn 3560  cop 3563   cuni 3772  dom cdm 4586  ran crn 4587   Fn wfn 5165  ontowfo 5168  1st c1st 6086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-fun 5172  df-fn 5173  df-fo 5176  df-1st 6088
This theorem is referenced by:  1stcof  6111  1stexg  6115  df1st2  6166  1stconst  6168  algrflem  6176  algrflemg  6177  suplocexprlemell  7633  suplocexprlem2b  7634  suplocexprlemlub  7644  upxp  12672  uptx  12674  cnmpt1st  12688
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