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Theorem 1stexg 5736
 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 2560 . 2 (A 𝑉A V)
2 fo1st 5726 . . . 4 1st :V–onto→V
3 fofn 5051 . . . 4 (1st :V–onto→V → 1st Fn V)
42, 3ax-mp 7 . . 3 1st Fn V
5 funfvex 5135 . . . 4 ((Fun 1st A dom 1st ) → (1stA) V)
65funfni 4942 . . 3 ((1st Fn V A V) → (1stA) V)
74, 6mpan 400 . 2 (A V → (1stA) V)
81, 7syl 14 1 (A 𝑉 → (1stA) V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1390  Vcvv 2551   Fn wfn 4840  –onto→wfo 4843  ‘cfv 4845  1st c1st 5707 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-1st 5709 This theorem is referenced by:  elxp7  5739  xpopth  5744  eqop  5745  2nd1st  5748  2ndrn  5751  releldm2  5753  reldm  5754  dfoprab3  5759  elopabi  5763  mpt2fvex  5771  dfmpt2  5786  cnvf1olem  5787  cnref1o  8357
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