Theorem mptexw 6139

Description: Weak version of mptex 5763 that holds without ax-coll 4133. If the domain and codomain of a function given by maps-to notation are sets, the function is a set. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Hypotheses

Ref	Expression
mptexw.1	|- A e. _V
mptexw.2	|- C e. _V
mptexw.3	|- A. x e. A B e. C

Assertion

Ref	Expression
mptexw	|- ( x e. A |-> B ) e. _V

Distinct variable groups:   x, A   x, C

Allowed substitution hint:   B( x)

Proof of Theorem mptexw

Step	Hyp	Ref	Expression
1		funmpt 5273	. 2 |- Fun ( x e. A |-> B )
2		mptexw.1	. . 3 |- A e. _V
3		eqid 2189	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
4	3	dmmptss 5143	. . 3 |- dom ( x e. A |-> B ) C_ A
5	2, 4	ssexi 4156	. 2 |- dom ( x e. A |-> B ) e. _V
6		mptexw.2	. . 3 |- C e. _V
7		mptexw.3	. . . 4 |- A. x e. A B e. C
8	3	rnmptss 5698	. . . 4 |- ( A. x e. A B e. C -> ran ( x e. A |-> B ) C_ C )
9	7, 8	ax-mp 5	. . 3 |- ran ( x e. A |-> B ) C_ C
10	6, 9	ssexi 4156	. 2 |- ran ( x e. A |-> B ) e. _V
11		funexw 6138	. 2 |- ( ( Fun ( x e. A |-> B ) /\ dom ( x e. A |-> B ) e. _V /\ ran ( x e. A |-> B ) e. _V ) -> ( x e. A |-> B ) e. _V )
12	1, 5, 10, 11	mp3an 1348	1 |- ( x e. A |-> B ) e. _V

Syntax hints:   e. wcel 2160   A.wral 2468   _Vcvv 2752   C_ wss 3144   |-> cmpt 4079   dom cdm 4644   ran crn 4645   Fun wfun 5229

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243

This theorem is referenced by: (None)