Theorem mptexw 7978

Description: Weak version of mptex 7244 that holds without ax-rep 5278. 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 6603	. 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
2		mptexw.1	. . 3 ⊢ 𝐴 ∈ V
3		eqid 2736	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	dmmptss 6260	. . 3 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴
5	2, 4	ssexi 5321	. 2 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V
6		mptexw.2	. . 3 ⊢ 𝐶 ∈ V
7		mptexw.3	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶
8	3	rnmptss 7142	. . . 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 5321	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V
11		funexw 7977	. 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 1462	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V

Syntax hints:   ∈ wcel 2107  ∀wral 3060  Vcvv 3479   ⊆ wss 3950   ↦ cmpt 5224  dom cdm 5684  ran crn 5685  Fun wfun 6554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-fun 6562  df-fn 6563  df-f 6564

This theorem is referenced by:  grpinvfval  18997  odfval  19551