Theorem mptexw 7654

Description: Weak version of mptex 6986 that holds without ax-rep 5190. 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 6393	. 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
2		mptexw.1	. . 3 ⊢ 𝐴 ∈ V
3		eqid 2821	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	dmmptss 6095	. . 3 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴
5	2, 4	ssexi 5226	. 2 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V
6		mptexw.2	. . 3 ⊢ 𝐶 ∈ V
7		mptexw.3	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶
8	3	rnmptss 6886	. . . 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 5226	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V
11		funexw 7653	. 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 1457	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V

Syntax hints:   ∈ wcel 2114  ∀wral 3138  Vcvv 3494   ⊆ wss 3936   ↦ cmpt 5146  dom cdm 5555  ran crn 5556  Fun wfun 6349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363

This theorem is referenced by:  grpinvfval  18142  odfval  18660