Theorem 1stexg 6374

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 2827	. 2 |- ( A e. V -> A e. _V )
2		fo1st 6364	. . . 4 |- 1st : _V -onto-> _V
3		fofn 5597	. . . 4 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
4	2, 3	ax-mp 5	. . 3 |- 1st Fn _V
5		funfvex 5692	. . . 4 |- ( ( Fun 1st /\ A e. dom 1st ) -> ( 1st ` A ) e. _V )
6	5	funfni 5463	. . 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 2205   _Vcvv 2815   Fn wfn 5352   -onto->wfo 5355   ` cfv 5357   1stc1st 6345

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fo 5363  df-fv 5365  df-1st 6347

This theorem is referenced by:  elxp7  6377  xpopth  6383  eqop  6384  2nd1st  6387  2ndrn  6390  releldm2  6392  reldm  6393  dfoprab3  6398  elopabi  6404  mpofvex  6414  dfmpo  6432  cnvf1olem  6433  cnvoprab  6443  f1od2  6444  disjxp1  6445  elmpom  6447  xpmapenlem  7115  cnref1o  10001  fsumcnv  12148  fprodcnv  12336  qredeu  12819  qnumval  12907  xpsff1o  13613  txbas  15249  txdis  15268  vtxvalg  16137  vtxex  16139  wlkelvv  16470  wlk2f  16472