Theorem 1stexg 6266

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 2785	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		fo1st 6256	. . . 4 ⊢ 1st :V–onto→V
3		fofn 5512	. . . 4 ⊢ (1st :V–onto→V → 1st Fn V)
4	2, 3	ax-mp 5	. . 3 ⊢ 1st Fn V
5		funfvex 5606	. . . 4 ⊢ ((Fun 1st ∧ 𝐴 ∈ dom 1st ) → (1st ‘𝐴) ∈ V)
6	5	funfni 5385	. . 3 ⊢ ((1st Fn V ∧ 𝐴 ∈ V) → (1st ‘𝐴) ∈ V)
7	4, 6	mpan 424	. 2 ⊢ (𝐴 ∈ V → (1st ‘𝐴) ∈ V)
8	1, 7	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → (1st ‘𝐴) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2177  Vcvv 2773   Fn wfn 5275  –onto→wfo 5278  ‘cfv 5280  1^st c1st 6237

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-fo 5286  df-fv 5288  df-1st 6239

This theorem is referenced by:  elxp7  6269  xpopth  6275  eqop  6276  2nd1st  6279  2ndrn  6282  releldm2  6284  reldm  6285  dfoprab3  6290  elopabi  6294  mpofvex  6304  dfmpo  6322  cnvf1olem  6323  cnvoprab  6333  f1od2  6334  disjxp1  6335  xpmapenlem  6961  cnref1o  9792  fsumcnv  11823  fprodcnv  12011  qredeu  12494  qnumval  12582  xpsff1o  13256  txbas  14805  txdis  14824  vtxvalg  15690  vtxex  15692