Theorem mptexw 7956

Description: Weak version of mptex 7220 that holds without ax-rep 5254. 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 ∈ V
mptexw.2	⊢ 𝐶 ∈ V
mptexw.3	⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶

Assertion

Ref	Expression
mptexw	⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem mptexw

Step	Hyp	Ref	Expression
1		funmpt 6579	. 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
2		mptexw.1	. . 3 ⊢ 𝐴 ∈ V
3		eqid 2736	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	dmmptss 6235	. . 3 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴
5	2, 4	ssexi 5297	. 2 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V
6		mptexw.2	. . 3 ⊢ 𝐶 ∈ V
7		mptexw.3	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶
8	3	rnmptss 7118	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐶)
9	7, 8	ax-mp 5	. . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐶
10	6, 9	ssexi 5297	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V
11		funexw 7955	. 2 ⊢ ((Fun (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
12	1, 5, 10, 11	mp3an 1463	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V

Syntax hints:   ∈ wcel 2109  ∀wral 3052  Vcvv 3464   ⊆ wss 3931   ↦ cmpt 5206  dom cdm 5659  ran crn 5660  Fun wfun 6530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-fun 6538  df-fn 6539  df-f 6540

This theorem is referenced by:  grpinvfval  18966  odfval  19518