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Theorem 1stexg 5715
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 5705 . . . 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 1374  Vcvv 2533   Fn wfn 4822  ontowfo 4825  cfv 4827  1st c1st 5686
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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3847  ax-pow 3899  ax-pr 3916  ax-un 4118
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  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 2897  df-in 2899  df-ss 2906  df-pw 3334  df-sn 3354  df-pr 3355  df-op 3357  df-uni 3553  df-br 3737  df-opab 3791  df-mpt 3792  df-id 4002  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 5688
This theorem is referenced by:  elxp7  5718  xpopth  5723  eqop  5724  2nd1st  5727  2ndrn  5730  releldm2  5732  reldm  5733  dfoprab3  5738  elopabi  5742  mpt2fvex  5750  dfmpt2  5765  cnvf1olem  5766
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