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Theorem 1stexg 5873
Description: Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
Assertion
Ref Expression
1stexg (𝐴𝑉 → (1st𝐴) ∈ V)

Proof of Theorem 1stexg
StepHypRef Expression
1 elex 2621 . 2 (𝐴𝑉𝐴 ∈ V)
2 fo1st 5863 . . . 4 1st :V–onto→V
3 fofn 5183 . . . 4 (1st :V–onto→V → 1st Fn V)
42, 3ax-mp 7 . . 3 1st Fn V
5 funfvex 5267 . . . 4 ((Fun 1st𝐴 ∈ dom 1st ) → (1st𝐴) ∈ V)
65funfni 5067 . . 3 ((1st Fn V ∧ 𝐴 ∈ V) → (1st𝐴) ∈ V)
74, 6mpan 415 . 2 (𝐴 ∈ V → (1st𝐴) ∈ V)
81, 7syl 14 1 (𝐴𝑉 → (1st𝐴) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1434  Vcvv 2612   Fn wfn 4964  ontowfo 4967  cfv 4969  1st c1st 5844
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-fo 4975  df-fv 4977  df-1st 5846
This theorem is referenced by:  elxp7  5876  xpopth  5881  eqop  5882  2nd1st  5885  2ndrn  5888  releldm2  5890  reldm  5891  dfoprab3  5896  elopabi  5900  mpt2fvex  5908  dfmpt2  5923  cnvf1olem  5924  cnvoprab  5934  f1od2  5935  xpmapenlem  6495  cnref1o  9028  qredeu  10859  qnumval  10943
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