Theorem 2ndexg 6171

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 2750	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		fo2nd 6161	. . . 4 ⊢ 2nd :V–onto→V
3		fofn 5442	. . . 4 ⊢ (2nd :V–onto→V → 2nd Fn V)
4	2, 3	ax-mp 5	. . 3 ⊢ 2nd Fn V
5		funfvex 5534	. . . 4 ⊢ ((Fun 2nd ∧ 𝐴 ∈ dom 2nd ) → (2nd ‘𝐴) ∈ V)
6	5	funfni 5318	. . 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 2148  Vcvv 2739   Fn wfn 5213  –onto→wfo 5216  ‘cfv 5218  2^nd c2nd 6142

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fo 5224  df-fv 5226  df-2nd 6144

This theorem is referenced by:  elxp7  6173  xpopth  6179  eqop  6180  op1steq  6182  2nd1st  6183  2ndrn  6186  dfoprab3  6194  elopabi  6198  mpofvex  6206  dfmpo  6226  cnvf1olem  6227  cnvoprab  6237  f1od2  6238  xpmapenlem  6851  cc2lem  7267  cnref1o  9652  fsumcnv  11447  fprodcnv  11635  qredeu  12099  qdenval  12188  xpsff1o  12773  txbas  13797  txdis  13816