Theorem 1stexg 5715

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 5705	. . . 4 ⊢ 1st :V–onto→V
3		fofn 5031	. . . 4 ⊢ (1st :V–onto→V → 1st Fn V)
4	2, 3	ax-mp 7	. . 3 ⊢ 1st Fn V
5		funfvex 5115	. . . 4 ⊢ ((Fun 1st ∧ A ∈ dom 1st ) → (1st ‘A) ∈ V)
6	5	funfni 4923	. . 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 1374  Vcvv 2533   Fn wfn 4822  –onto→wfo 4825  ‘cfv 4827  1^st c1st 5686

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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3847  ax-pow 3899  ax-pr 3916  ax-un 4118

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  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 2897  df-in 2899  df-ss 2906  df-pw 3334  df-sn 3354  df-pr 3355  df-op 3357  df-uni 3553  df-br 3737  df-opab 3791  df-mpt 3792  df-id 4002  df-xp 4276  df-rel 4277  df-cnv 4278  df-co 4279  df-dm 4280  df-rn 4281  df-iota 4792  df-fun 4829  df-fn 4830  df-f 4831  df-fo 4833  df-fv 4835  df-1st 5688

This theorem is referenced by:  elxp7  5718  xpopth  5723  eqop  5724  2nd1st  5727  2ndrn  5730  releldm2  5732  reldm  5733  dfoprab3  5738  elopabi  5742  mpt2fvex  5750  dfmpt2  5765  cnvf1olem  5766