Theorem funexw 6112

Description: Weak version of funex 5739 that holds without ax-coll 4118. 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 4740	. . 3 |- ( ( dom F e. B /\ ran F e. C ) -> ( dom F X. ran F ) e. _V )
2	1	3adant1 1015	. 2 |- ( ( Fun F /\ dom F e. B /\ ran F e. C ) -> ( dom F X. ran F ) e. _V )
3		funrel 5233	. . . 4 |- ( Fun F -> Rel F )
4		relssdmrn 5149	. . . 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 1018	. 2 |- ( ( Fun F /\ dom F e. B /\ ran F e. C ) -> F C_ ( dom F X. ran F ) )
7	2, 6	ssexd 4143	1 |- ( ( Fun F /\ dom F e. B /\ ran F e. C ) -> F e. _V )

Syntax hints:   -> wi 4   /\ w3a 978   e. wcel 2148   _Vcvv 2737   C_ wss 3129   X. cxp 4624   dom cdm 4626   ran crn 4627   Rel wrel 4631   Fun wfun 5210

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633  df-cnv 4634  df-dm 4636  df-rn 4637  df-fun 5218

This theorem is referenced by:  mptexw  6113  mpoexw  6213