Theorem mptexw 6256

Description: Weak version of mptex 5864 that holds without ax-coll 4198. 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 5355	. 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
2		mptexw.1	. . 3 ⊢ 𝐴 ∈ V
3		eqid 2229	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	dmmptss 5224	. . 3 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴
5	2, 4	ssexi 4221	. 2 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V
6		mptexw.2	. . 3 ⊢ 𝐶 ∈ V
7		mptexw.3	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶
8	3	rnmptss 5795	. . . 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 4221	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V
11		funexw 6255	. 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 1371	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V

Syntax hints:   ∈ wcel 2200  ∀wral 2508  Vcvv 2799   ⊆ wss 3197   ↦ cmpt 4144  dom cdm 4718  ran crn 4719  Fun wfun 5311

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fv 5325

This theorem is referenced by: (None)