Theorem 1stexg 6222

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 2771	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		fo1st 6212	. . . 4 ⊢ 1st :V–onto→V
3		fofn 5479	. . . 4 ⊢ (1st :V–onto→V → 1st Fn V)
4	2, 3	ax-mp 5	. . 3 ⊢ 1st Fn V
5		funfvex 5572	. . . 4 ⊢ ((Fun 1st ∧ 𝐴 ∈ dom 1st ) → (1st ‘𝐴) ∈ V)
6	5	funfni 5355	. . 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 2164  Vcvv 2760   Fn wfn 5250  –onto→wfo 5253  ‘cfv 5255  1^st c1st 6193

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6195

This theorem is referenced by:  elxp7  6225  xpopth  6231  eqop  6232  2nd1st  6235  2ndrn  6238  releldm2  6240  reldm  6241  dfoprab3  6246  elopabi  6250  mpofvex  6260  dfmpo  6278  cnvf1olem  6279  cnvoprab  6289  f1od2  6290  disjxp1  6291  xpmapenlem  6907  cnref1o  9719  fsumcnv  11583  fprodcnv  11771  qredeu  12238  qnumval  12326  xpsff1o  12935  txbas  14437  txdis  14456