Theorem 2ndexg 6396

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 2833	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		fo2nd 6386	. . . 4 ⊢ 2nd :V–onto→V
3		fofn 5615	. . . 4 ⊢ (2nd :V–onto→V → 2nd Fn V)
4	2, 3	ax-mp 5	. . 3 ⊢ 2nd Fn V
5		funfvex 5710	. . . 4 ⊢ ((Fun 2nd ∧ 𝐴 ∈ dom 2nd ) → (2nd ‘𝐴) ∈ V)
6	5	funfni 5481	. . 3 ⊢ ((2nd Fn V ∧ 𝐴 ∈ V) → (2nd ‘𝐴) ∈ V)
7	4, 6	mpan 428	. 2 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) ∈ V)
8	1, 7	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → (2nd ‘𝐴) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2209  Vcvv 2821   Fn wfn 5370  –onto→wfo 5373  ‘cfv 5375  2^nd c2nd 6367

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fo 5381  df-fv 5383  df-2nd 6369

This theorem is referenced by:  elxp7  6398  xpopth  6404  eqop  6405  op1steq  6407  2nd1st  6408  2ndrn  6411  dfoprab3  6419  elopabi  6425  mpofvex  6435  dfmpo  6453  cnvf1olem  6454  cnvoprab  6464  f1od2  6465  elmpom  6468  xpmapenlem  7143  cc2lem  7626  cnref1o  10034  fsumcnv  12187  fprodcnv  12375  qredeu  12858  qdenval  12947  xpsff1o  13653  txbas  15342  txdis  15361  iedgvalg  16241  iedgex  16243  edgvalg  16283  wlkelvv  16573  wlk2f  16575