Theorem funexw 6257

Description: Weak version of funex 5862 that holds without ax-coll 4199. If the domain and codomain of a function exist, so does the function. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Assertion

Ref	Expression
funexw	|- ( ( Fun F /\ dom F e. B /\ ran F e. C ) -> F e. _V )

Proof of Theorem funexw

Step	Hyp	Ref	Expression
1		xpexg 4833	. . 3 |- ( ( dom F e. B /\ ran F e. C ) -> ( dom F X. ran F ) e. _V )
2	1	3adant1 1039	. 2 |- ( ( Fun F /\ dom F e. B /\ ran F e. C ) -> ( dom F X. ran F ) e. _V )
3		funrel 5335	. . . 4 |- ( Fun F -> Rel F )
4		relssdmrn 5249	. . . 4 |- ( Rel F -> F C_ ( dom F X. ran F ) )
5	3, 4	syl 14	. . 3 |- ( Fun F -> F C_ ( dom F X. ran F ) )
6	5	3ad2ant1 1042	. 2 |- ( ( Fun F /\ dom F e. B /\ ran F e. C ) -> F C_ ( dom F X. ran F ) )
7	2, 6	ssexd 4224	1 |- ( ( Fun F /\ dom F e. B /\ ran F e. C ) -> F e. _V )

Syntax hints:   -> wi 4   /\ w3a 1002   e. wcel 2200   _Vcvv 2799   C_ wss 3197   X. cxp 4717   dom cdm 4719   ran crn 4720   Rel wrel 4724   Fun wfun 5312

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727  df-dm 4729  df-rn 4730  df-fun 5320

This theorem is referenced by:  mptexw  6258  mpoexw  6359