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Theorem 1stexg 5714
Description: Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
Assertion
Ref Expression
1stexg (A 𝑉 → (1stA) V)

Proof of Theorem 1stexg
StepHypRef Expression
1 elex 2542 . 2 (A 𝑉A V)
2 fo1st 5704 . . . 4 1st :V–onto→V
3 fofn 5031 . . . 4 (1st :V–onto→V → 1st Fn V)
42, 3ax-mp 7 . . 3 1st Fn V
5 funfvex 5115 . . . 4 ((Fun 1st A dom 1st ) → (1stA) V)
65funfni 4923 . . 3 ((1st Fn V A V) → (1stA) V)
74, 6mpan 402 . 2 (A V → (1stA) V)
81, 7syl 14 1 (A 𝑉 → (1stA) V)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1375  Vcvv 2534   Fn wfn 4822  ontowfo 4825  cfv 4827  1st c1st 5685
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-sep 3848  ax-pow 3900  ax-pr 3917  ax-un 4118
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-eu 1886  df-mo 1887  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-v 2536  df-sbc 2741  df-un 2898  df-in 2900  df-ss 2907  df-pw 3335  df-sn 3355  df-pr 3356  df-op 3358  df-uni 3554  df-br 3738  df-opab 3792  df-mpt 3793  df-id 4003  df-xp 4276  df-rel 4277  df-cnv 4278  df-co 4279  df-dm 4280  df-rn 4281  df-iota 4792  df-fun 4829  df-fn 4830  df-f 4831  df-fo 4833  df-fv 4835  df-1st 5687
This theorem is referenced by:  elxp7  5717  xpopth  5722  eqop  5723  2nd1st  5726  2ndrn  5729  releldm2  5731  reldm  5732  dfoprab3  5737  elopabi  5741  mpt2fvex  5749  dfmpt2  5764  cnvf1olem  5765
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