Theorem 1stexg 6252

Description: Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)

Assertion

Ref	Expression
1stexg	⊢ (𝐴 ∈ 𝑉 → (1st ‘𝐴) ∈ V)

Proof of Theorem 1stexg

Step	Hyp	Ref	Expression
1		elex 2782	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		fo1st 6242	. . . 4 ⊢ 1st :V–onto→V
3		fofn 5499	. . . 4 ⊢ (1st :V–onto→V → 1st Fn V)
4	2, 3	ax-mp 5	. . 3 ⊢ 1st Fn V
5		funfvex 5592	. . . 4 ⊢ ((Fun 1st ∧ 𝐴 ∈ dom 1st ) → (1st ‘𝐴) ∈ V)
6	5	funfni 5375	. . 3 ⊢ ((1st Fn V ∧ 𝐴 ∈ V) → (1st ‘𝐴) ∈ V)
7	4, 6	mpan 424	. 2 ⊢ (𝐴 ∈ V → (1st ‘𝐴) ∈ V)
8	1, 7	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → (1st ‘𝐴) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2175  Vcvv 2771   Fn wfn 5265  –onto→wfo 5268  ‘cfv 5270  1^st c1st 6223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-fo 5276  df-fv 5278  df-1st 6225

This theorem is referenced by:  elxp7  6255  xpopth  6261  eqop  6262  2nd1st  6265  2ndrn  6268  releldm2  6270  reldm  6271  dfoprab3  6276  elopabi  6280  mpofvex  6290  dfmpo  6308  cnvf1olem  6309  cnvoprab  6319  f1od2  6320  disjxp1  6321  xpmapenlem  6945  cnref1o  9771  fsumcnv  11690  fprodcnv  11878  qredeu  12361  qnumval  12449  xpsff1o  13123  txbas  14672  txdis  14691  vtxvalg  15557