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Theorem 1stexg 5683
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 2541 . 2 (A 𝑉A V)
2 fo1st 5673 . . . 4 1st :V–onto→V
3 fofn 5000 . . . 4 (1st :V–onto→V → 1st Fn V)
42, 3ax-mp 7 . . 3 1st Fn V
5 funfvex 5084 . . . 4 ((Fun 1st A dom 1st ) → (1stA) V)
65funfni 4892 . . 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 2533   Fn wfn 4791  ontowfo 4794  cfv 4796  1st c1st 5654
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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  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 2004  ax-sep 3827  ax-pow 3879  ax-pr 3896  ax-un 4093
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-eu 1884  df-mo 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-sbc 2740  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336  df-uni 3533  df-br 3717  df-opab 3771  df-mpt 3772  df-id 3983  df-xp 4244  df-rel 4245  df-cnv 4246  df-co 4247  df-dm 4248  df-rn 4249  df-iota 4761  df-fun 4798  df-fn 4799  df-f 4800  df-fo 4802  df-fv 4804  df-1st 5656
This theorem is referenced by:  elxp7  5686  xpopth  5691  eqop  5692  2nd1st  5695  2ndrn  5698  releldm2  5700  reldm  5701  dfoprab3  5706  elopabi  5710  mpt2fvex  5718  dfmpt2  5733  cnvf1olem  5734
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