Theorem 1stexg 5736

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 2560	. 2 ⊢ (A ∈ 𝑉 → A ∈ V)
2		fo1st 5726	. . . 4 ⊢ 1st :V–onto→V
3		fofn 5051	. . . 4 ⊢ (1st :V–onto→V → 1st Fn V)
4	2, 3	ax-mp 7	. . 3 ⊢ 1st Fn V
5		funfvex 5135	. . . 4 ⊢ ((Fun 1st ∧ A ∈ dom 1st ) → (1st ‘A) ∈ V)
6	5	funfni 4942	. . 3 ⊢ ((1st Fn V ∧ A ∈ V) → (1st ‘A) ∈ V)
7	4, 6	mpan 400	. 2 ⊢ (A ∈ V → (1st ‘A) ∈ V)
8	1, 7	syl 14	1 ⊢ (A ∈ 𝑉 → (1st ‘A) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 1390  Vcvv 2551   Fn wfn 4840  –onto→wfo 4843  ‘cfv 4845  1^st c1st 5707

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-1st 5709

This theorem is referenced by:  elxp7  5739  xpopth  5744  eqop  5745  2nd1st  5748  2ndrn  5751  releldm2  5753  reldm  5754  dfoprab3  5759  elopabi  5763  mpt2fvex  5771  dfmpt2  5786  cnvf1olem  5787  cnref1o  8337