Theorem 1stexg 6276

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 2788	. 2 |- ( A e. V -> A e. _V )
2		fo1st 6266	. . . 4 |- 1st : _V -onto-> _V
3		fofn 5522	. . . 4 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
4	2, 3	ax-mp 5	. . 3 |- 1st Fn _V
5		funfvex 5616	. . . 4 |- ( ( Fun 1st /\ A e. dom 1st ) -> ( 1st ` A ) e. _V )
6	5	funfni 5395	. . 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 2178   _Vcvv 2776   Fn wfn 5285   -onto->wfo 5288   ` cfv 5290   1stc1st 6247

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 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

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 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fo 5296  df-fv 5298  df-1st 6249

This theorem is referenced by:  elxp7  6279  xpopth  6285  eqop  6286  2nd1st  6289  2ndrn  6292  releldm2  6294  reldm  6295  dfoprab3  6300  elopabi  6304  mpofvex  6314  dfmpo  6332  cnvf1olem  6333  cnvoprab  6343  f1od2  6344  disjxp1  6345  xpmapenlem  6971  cnref1o  9807  fsumcnv  11863  fprodcnv  12051  qredeu  12534  qnumval  12622  xpsff1o  13296  txbas  14845  txdis  14864  vtxvalg  15730  vtxex  15732