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Theorem fo1st 6136
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 2733 . . . . . 6 𝑥 ∈ V
21snex 4171 . . . . 5 {𝑥} ∈ V
32dmex 4877 . . . 4 dom {𝑥} ∈ V
43uniex 4422 . . 3 dom {𝑥} ∈ V
5 df-1st 6119 . . 3 1st = (𝑥 ∈ V ↦ dom {𝑥})
64, 5fnmpti 5326 . 2 1st Fn V
75rnmpt 4859 . . 3 ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = dom {𝑥}}
8 vex 2733 . . . . 5 𝑦 ∈ V
98, 8opex 4214 . . . . . 6 𝑦, 𝑦⟩ ∈ V
108, 8op1sta 5092 . . . . . . 7 dom {⟨𝑦, 𝑦⟩} = 𝑦
1110eqcomi 2174 . . . . . 6 𝑦 = dom {⟨𝑦, 𝑦⟩}
12 sneq 3594 . . . . . . . . . 10 (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
1312dmeqd 4813 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
1413unieqd 3807 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
1514eqeq2d 2182 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑦⟩ → (𝑦 = dom {𝑥} ↔ 𝑦 = dom {⟨𝑦, 𝑦⟩}))
1615rspcev 2834 . . . . . 6 ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = dom {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = dom {𝑥})
179, 11, 16mp2an 424 . . . . 5 𝑥 ∈ V 𝑦 = dom {𝑥}
188, 172th 173 . . . 4 (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = dom {𝑥})
1918abbi2i 2285 . . 3 V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = dom {𝑥}}
207, 19eqtr4i 2194 . 2 ran 1st = V
21 df-fo 5204 . 2 (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
226, 20, 21mpbir2an 937 1 1st :V–onto→V
Colors of variables: wff set class
Syntax hints:   = wceq 1348  wcel 2141  {cab 2156  wrex 2449  Vcvv 2730  {csn 3583  cop 3586   cuni 3796  dom cdm 4611  ran crn 4612   Fn wfn 5193  ontowfo 5196  1st c1st 6117
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-fun 5200  df-fn 5201  df-fo 5204  df-1st 6119
This theorem is referenced by:  1stcof  6142  1stexg  6146  df1st2  6198  1stconst  6200  algrflem  6208  algrflemg  6209  suplocexprlemell  7675  suplocexprlem2b  7676  suplocexprlemlub  7686  upxp  13066  uptx  13068  cnmpt1st  13082
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