Theorem 1stexg 6401

Description: Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)

Assertion

Ref	Expression
1stexg	|- ( A e. V -> ( 1st ` A ) e. _V )

Proof of Theorem 1stexg

Step	Hyp	Ref	Expression
1		elex 2833	. 2 |- ( A e. V -> A e. _V )
2		fo1st 6391	. . . 4 |- 1st : _V -onto-> _V
3		fofn 5617	. . . 4 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
4	2, 3	ax-mp 5	. . 3 |- 1st Fn _V
5		funfvex 5712	. . . 4 |- ( ( Fun 1st /\ A e. dom 1st ) -> ( 1st ` A ) e. _V )
6	5	funfni 5483	. . 3 |- ( ( 1st Fn _V /\ A e. _V ) -> ( 1st ` A ) e. _V )
7	4, 6	mpan 428	. 2 |- ( A e. _V -> ( 1st ` A ) e. _V )
8	1, 7	syl 14	1 |- ( A e. V -> ( 1st ` A ) e. _V )

This proof depends on syntax axioms:   -> wi 4   e. wcel 2209   _Vcvv 2821   Fn wfn 5372   -onto->wfo 5375   ` cfv 5377   1stc1st 6372

This proof depends on 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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fo 5383  df-fv 5385  df-1st 6374

This theorem is used by:  elxp7  6404  xpopth  6410  eqop  6411  2nd1st  6414  2ndrn  6417  releldm2  6419  reldm  6420  dfoprab3  6425  elopabi  6431  mpofvex  6441  dfmpo  6459  cnvf1olem  6460  cnvoprab  6470  f1od2  6471  disjxp1  6472  elmpom  6474  xpmapenlem  7149  cnref1o  10051  fsumcnv  12204  fprodcnv  12392  qredeu  12875  qnumval  12963  xpsff1o  13670  txbas  15359  txdis  15378  vtxvalg  16257  vtxex  16259  wlkelvv  16590  wlk2f  16592