Theorem 1stexg 6253

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 2783	. 2 |- ( A e. V -> A e. _V )
2		fo1st 6243	. . . 4 |- 1st : _V -onto-> _V
3		fofn 5500	. . . 4 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
4	2, 3	ax-mp 5	. . 3 |- 1st Fn _V
5		funfvex 5593	. . . 4 |- ( ( Fun 1st /\ A e. dom 1st ) -> ( 1st ` A ) e. _V )
6	5	funfni 5376	. . 3 |- ( ( 1st Fn _V /\ A e. _V ) -> ( 1st ` A ) e. _V )
7	4, 6	mpan 424	. 2 |- ( A e. _V -> ( 1st ` A ) e. _V )
8	1, 7	syl 14	1 |- ( A e. V -> ( 1st ` A ) e. _V )

Syntax hints:   -> wi 4   e. wcel 2176   _Vcvv 2772   Fn wfn 5266   -onto->wfo 5269   ` cfv 5271   1stc1st 6224

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6226

This theorem is referenced by:  elxp7  6256  xpopth  6262  eqop  6263  2nd1st  6266  2ndrn  6269  releldm2  6271  reldm  6272  dfoprab3  6277  elopabi  6281  mpofvex  6291  dfmpo  6309  cnvf1olem  6310  cnvoprab  6320  f1od2  6321  disjxp1  6322  xpmapenlem  6946  cnref1o  9772  fsumcnv  11748  fprodcnv  11936  qredeu  12419  qnumval  12507  xpsff1o  13181  txbas  14730  txdis  14749  vtxvalg  15615