Theorem 1stexg 6311

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 2811	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		fo1st 6301	. . . 4 ⊢ 1st :V–onto→V
3		fofn 5549	. . . 4 ⊢ (1st :V–onto→V → 1st Fn V)
4	2, 3	ax-mp 5	. . 3 ⊢ 1st Fn V
5		funfvex 5643	. . . 4 ⊢ ((Fun 1st ∧ 𝐴 ∈ dom 1st ) → (1st ‘𝐴) ∈ V)
6	5	funfni 5422	. . 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 2200  Vcvv 2799   Fn wfn 5312  –onto→wfo 5315  ‘cfv 5317  1^st c1st 6282

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fo 5323  df-fv 5325  df-1st 6284

This theorem is referenced by:  elxp7  6314  xpopth  6320  eqop  6321  2nd1st  6324  2ndrn  6327  releldm2  6329  reldm  6330  dfoprab3  6335  elopabi  6339  mpofvex  6349  dfmpo  6367  cnvf1olem  6368  cnvoprab  6378  f1od2  6379  disjxp1  6380  xpmapenlem  7006  cnref1o  9842  fsumcnv  11943  fprodcnv  12131  qredeu  12614  qnumval  12702  xpsff1o  13377  txbas  14926  txdis  14945  vtxvalg  15811  vtxex  15813