Theorem mptexw 6116

Description: Weak version of mptex 5744 that holds without ax-coll 4120. 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 5256	. 2 |- Fun ( x e. A |-> B )
2		mptexw.1	. . 3 |- A e. _V
3		eqid 2177	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
4	3	dmmptss 5127	. . 3 |- dom ( x e. A |-> B ) C_ A
5	2, 4	ssexi 4143	. 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 5679	. . . 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 4143	. 2 |- ran ( x e. A |-> B ) e. _V
11		funexw 6115	. 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 1337	1 |- ( x e. A |-> B ) e. _V

Syntax hints:   e. wcel 2148   A.wral 2455   _Vcvv 2739   C_ wss 3131   |-> cmpt 4066   dom cdm 4628   ran crn 4629   Fun wfun 5212

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 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

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-rab 2464  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226

This theorem is referenced by: (None)