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Theorem 2ndexg 5877
Description: Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
Assertion
Ref Expression
2ndexg (𝐴𝑉 → (2nd𝐴) ∈ V)

Proof of Theorem 2ndexg
StepHypRef Expression
1 elex 2623 . 2 (𝐴𝑉𝐴 ∈ V)
2 fo2nd 5867 . . . 4 2nd :V–onto→V
3 fofn 5186 . . . 4 (2nd :V–onto→V → 2nd Fn V)
42, 3ax-mp 7 . . 3 2nd Fn V
5 funfvex 5270 . . . 4 ((Fun 2nd𝐴 ∈ dom 2nd ) → (2nd𝐴) ∈ V)
65funfni 5070 . . 3 ((2nd Fn V ∧ 𝐴 ∈ V) → (2nd𝐴) ∈ V)
74, 6mpan 415 . 2 (𝐴 ∈ V → (2nd𝐴) ∈ V)
81, 7syl 14 1 (𝐴𝑉 → (2nd𝐴) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1436  Vcvv 2614   Fn wfn 4967  ontowfo 4970  cfv 4972  2nd c2nd 5848
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-sbc 2829  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-br 3815  df-opab 3869  df-mpt 3870  df-id 4087  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-fo 4978  df-fv 4980  df-2nd 5850
This theorem is referenced by:  elxp7  5879  xpopth  5884  eqop  5885  op1steq  5887  2nd1st  5888  2ndrn  5891  dfoprab3  5899  elopabi  5903  mpt2fvex  5911  dfmpt2  5926  cnvf1olem  5927  cnvoprab  5937  f1od2  5938  xpmapenlem  6498  djur  6678  cnref1o  9042  qredeu  10873  qdenval  10958
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