Theorem 1stexg 5683

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 2541	. 2 ⊢ (A ∈ 𝑉 → A ∈ V)
2		fo1st 5673	. . . 4 ⊢ 1st :V–onto→V
3		fofn 5000	. . . 4 ⊢ (1st :V–onto→V → 1st Fn V)
4	2, 3	ax-mp 7	. . 3 ⊢ 1st Fn V
5		funfvex 5084	. . . 4 ⊢ ((Fun 1st ∧ A ∈ dom 1st ) → (1st ‘A) ∈ V)
6	5	funfni 4892	. . 3 ⊢ ((1st Fn V ∧ A ∈ V) → (1st ‘A) ∈ V)
7	4, 6	mpan 402	. 2 ⊢ (A ∈ V → (1st ‘A) ∈ V)
8	1, 7	syl 14	1 ⊢ (A ∈ 𝑉 → (1st ‘A) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 1375  Vcvv 2533   Fn wfn 4791  –onto→wfo 4794  ‘cfv 4796  1^st c1st 5654

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-sep 3827  ax-pow 3879  ax-pr 3896  ax-un 4093

This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-eu 1884  df-mo 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-sbc 2740  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336  df-uni 3533  df-br 3717  df-opab 3771  df-mpt 3772  df-id 3983  df-xp 4244  df-rel 4245  df-cnv 4246  df-co 4247  df-dm 4248  df-rn 4249  df-iota 4761  df-fun 4798  df-fn 4799  df-f 4800  df-fo 4802  df-fv 4804  df-1st 5656

This theorem is referenced by:  elxp7  5686  xpopth  5691  eqop  5692  2nd1st  5695  2ndrn  5698  releldm2  5700  reldm  5701  dfoprab3  5706  elopabi  5710  mpt2fvex  5718  dfmpt2  5733  cnvf1olem  5734