Theorem mptexw 7769

Description: Weak version of mptex 7081 that holds without ax-rep 5205. 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 6456	. 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
2		mptexw.1	. . 3 ⊢ 𝐴 ∈ V
3		eqid 2738	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	dmmptss 6133	. . 3 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴
5	2, 4	ssexi 5241	. 2 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V
6		mptexw.2	. . 3 ⊢ 𝐶 ∈ V
7		mptexw.3	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶
8	3	rnmptss 6978	. . . 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 5241	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V
11		funexw 7768	. 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 1459	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V

Syntax hints:   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ⊆ wss 3883   ↦ cmpt 5153  dom cdm 5580  ran crn 5581  Fun wfun 6412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-fun 6420  df-fn 6421  df-f 6422

This theorem is referenced by:  grpinvfval  18533  odfval  19055