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Theorem fo1st 6021
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 2661 . . . . . 6 𝑥 ∈ V
21snex 4077 . . . . 5 {𝑥} ∈ V
32dmex 4773 . . . 4 dom {𝑥} ∈ V
43uniex 4327 . . 3 dom {𝑥} ∈ V
5 df-1st 6004 . . 3 1st = (𝑥 ∈ V ↦ dom {𝑥})
64, 5fnmpti 5219 . 2 1st Fn V
75rnmpt 4755 . . 3 ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = dom {𝑥}}
8 vex 2661 . . . . 5 𝑦 ∈ V
98, 8opex 4119 . . . . . 6 𝑦, 𝑦⟩ ∈ V
108, 8op1sta 4988 . . . . . . 7 dom {⟨𝑦, 𝑦⟩} = 𝑦
1110eqcomi 2119 . . . . . 6 𝑦 = dom {⟨𝑦, 𝑦⟩}
12 sneq 3506 . . . . . . . . . 10 (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
1312dmeqd 4709 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
1413unieqd 3715 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
1514eqeq2d 2127 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑦⟩ → (𝑦 = dom {𝑥} ↔ 𝑦 = dom {⟨𝑦, 𝑦⟩}))
1615rspcev 2761 . . . . . 6 ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = dom {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = dom {𝑥})
179, 11, 16mp2an 420 . . . . 5 𝑥 ∈ V 𝑦 = dom {𝑥}
188, 172th 173 . . . 4 (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = dom {𝑥})
1918abbi2i 2230 . . 3 V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = dom {𝑥}}
207, 19eqtr4i 2139 . 2 ran 1st = V
21 df-fo 5097 . 2 (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
226, 20, 21mpbir2an 909 1 1st :V–onto→V
Colors of variables: wff set class
Syntax hints:   = wceq 1314  wcel 1463  {cab 2101  wrex 2392  Vcvv 2658  {csn 3495  cop 3498   cuni 3704  dom cdm 4507  ran crn 4508   Fn wfn 5086  ontowfo 5089  1st c1st 6002
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-fun 5093  df-fn 5094  df-fo 5097  df-1st 6004
This theorem is referenced by:  1stcof  6027  1stexg  6031  df1st2  6082  1stconst  6084  algrflem  6092  algrflemg  6093  suplocexprlemell  7485  suplocexprlem2b  7486  suplocexprlemlub  7496  upxp  12336  uptx  12338  cnmpt1st  12352
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