Theorem mptexw 6346

Description: Weak version of mptex 5946 that holds without ax-coll 4248. 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 5418	. 2 |- Fun ( x e. A |-> B )
2		mptexw.1	. . 3 |- A e. _V
3		eqid 2238	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
4	3	dmmptss 5287	. . 3 |- dom ( x e. A |-> B ) C_ A
5	2, 4	ssexi 4273	. 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 5872	. . . 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 4273	. 2 |- ran ( x e. A |-> B ) e. _V
11		funexw 6345	. 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 1378	1 |- ( x e. A |-> B ) e. _V

This proof depends on syntax axioms:   e. wcel 2209   A.wral 2528   _Vcvv 2821   C_ wss 3220   |-> cmpt 4193   dom cdm 4776   ran crn 4777   Fun wfun 5374

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388

This theorem is used by: (None)