Theorem 2ndexg 6361

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

Assertion

Ref	Expression
2ndexg	⊢ (𝐴 ∈ 𝑉 → (2nd ‘𝐴) ∈ V)

Proof of Theorem 2ndexg

Step	Hyp	Ref	Expression
1		elex 2824	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		fo2nd 6351	. . . 4 ⊢ 2nd :V–onto→V
3		fofn 5591	. . . 4 ⊢ (2nd :V–onto→V → 2nd Fn V)
4	2, 3	ax-mp 5	. . 3 ⊢ 2nd Fn V
5		funfvex 5686	. . . 4 ⊢ ((Fun 2nd ∧ 𝐴 ∈ dom 2nd ) → (2nd ‘𝐴) ∈ V)
6	5	funfni 5457	. . 3 ⊢ ((2nd Fn V ∧ 𝐴 ∈ V) → (2nd ‘𝐴) ∈ V)
7	4, 6	mpan 424	. 2 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) ∈ V)
8	1, 7	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → (2nd ‘𝐴) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2203  Vcvv 2812   Fn wfn 5346  –onto→wfo 5349  ‘cfv 5351  2^nd c2nd 6332

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-sbc 3042  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fo 5357  df-fv 5359  df-2nd 6334

This theorem is referenced by:  elxp7  6363  xpopth  6369  eqop  6370  op1steq  6372  2nd1st  6373  2ndrn  6376  dfoprab3  6384  elopabi  6390  mpofvex  6400  dfmpo  6418  cnvf1olem  6419  cnvoprab  6429  f1od2  6430  elmpom  6433  xpmapenlem  7101  cc2lem  7576  cnref1o  9979  fsumcnv  12116  fprodcnv  12304  qredeu  12787  qdenval  12876  xpsff1o  13551  txbas  15110  txdis  15129  iedgvalg  15999  iedgex  16001  edgvalg  16041  wlkelvv  16331  wlk2f  16333