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Theorem fo1st 5910
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 2622 . . . . . 6 𝑥 ∈ V
21snex 4011 . . . . 5 {𝑥} ∈ V
32dmex 4687 . . . 4 dom {𝑥} ∈ V
43uniex 4255 . . 3 dom {𝑥} ∈ V
5 df-1st 5893 . . 3 1st = (𝑥 ∈ V ↦ dom {𝑥})
64, 5fnmpti 5128 . 2 1st Fn V
75rnmpt 4671 . . 3 ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = dom {𝑥}}
8 vex 2622 . . . . 5 𝑦 ∈ V
98, 8opex 4047 . . . . . 6 𝑦, 𝑦⟩ ∈ V
108, 8op1sta 4899 . . . . . . 7 dom {⟨𝑦, 𝑦⟩} = 𝑦
1110eqcomi 2092 . . . . . 6 𝑦 = dom {⟨𝑦, 𝑦⟩}
12 sneq 3452 . . . . . . . . . 10 (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
1312dmeqd 4626 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
1413unieqd 3659 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
1514eqeq2d 2099 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑦⟩ → (𝑦 = dom {𝑥} ↔ 𝑦 = dom {⟨𝑦, 𝑦⟩}))
1615rspcev 2722 . . . . . 6 ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = dom {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = dom {𝑥})
179, 11, 16mp2an 417 . . . . 5 𝑥 ∈ V 𝑦 = dom {𝑥}
188, 172th 172 . . . 4 (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = dom {𝑥})
1918abbi2i 2202 . . 3 V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = dom {𝑥}}
207, 19eqtr4i 2111 . 2 ran 1st = V
21 df-fo 5008 . 2 (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
226, 20, 21mpbir2an 888 1 1st :V–onto→V
Colors of variables: wff set class
Syntax hints:   = wceq 1289  wcel 1438  {cab 2074  wrex 2360  Vcvv 2619  {csn 3441  cop 3444   cuni 3648  dom cdm 4428  ran crn 4429   Fn wfn 4997  ontowfo 5000  1st c1st 5891
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-fun 5004  df-fn 5005  df-fo 5008  df-1st 5893
This theorem is referenced by:  1stcof  5916  1stexg  5920  df1st2  5966  1stconst  5968  algrflem  5976  algrflemg  5977
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