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Theorem 1stexg 5794
 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 2566 . 2 (𝐴𝑉𝐴 ∈ V)
2 fo1st 5784 . . . 4 1st :V–onto→V
3 fofn 5108 . . . 4 (1st :V–onto→V → 1st Fn V)
42, 3ax-mp 7 . . 3 1st Fn V
5 funfvex 5192 . . . 4 ((Fun 1st𝐴 ∈ dom 1st ) → (1st𝐴) ∈ V)
65funfni 4999 . . 3 ((1st Fn V ∧ 𝐴 ∈ V) → (1st𝐴) ∈ V)
74, 6mpan 400 . 2 (𝐴 ∈ V → (1st𝐴) ∈ V)
81, 7syl 14 1 (𝐴𝑉 → (1st𝐴) ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1393  Vcvv 2557   Fn wfn 4897  –onto→wfo 4900  ‘cfv 4902  1st c1st 5765 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908  df-fv 4910  df-1st 5767 This theorem is referenced by:  elxp7  5797  xpopth  5802  eqop  5803  2nd1st  5806  2ndrn  5809  releldm2  5811  reldm  5812  dfoprab3  5817  elopabi  5821  mpt2fvex  5829  dfmpt2  5844  cnvf1olem  5845  cnref1o  8582
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