Theorem mptssid 46346

Description: The mapping operation expressed with its actual domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
mptssid.1	⊢ Ⅎ𝑥𝐴
mptssid.2	⊢ 𝐶 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}

Assertion

Ref	Expression
mptssid	⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐵)

Proof of Theorem mptssid

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqvisset 3470	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → 𝐵 ∈ V)
2	1	anim2i 629	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V))
3		rabid 3432	. . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V))
4	2, 3	sylibr 237	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V})
5		mptssid.2	. . . . . 6 ⊢ 𝐶 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
6	4, 5	eleqtrrdi 2871	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝐶)
7		simpr 490	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
8	6, 7	jca 521	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵))
9		mptssid.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
10	9	ssrab2f 46225	. . . . . . 7 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ 𝐴
11	5, 10	eqsstri 3954	. . . . . 6 ⊢ 𝐶 ⊆ 𝐴
12	11	sseli 3904	. . . . 5 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴)
13	12	anim1i 627	. . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
14	8, 13	impbii 212	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵))
15	14	opabbii 5171	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵)}
16		df-mpt 5186	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
17		df-mpt 5186	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵)}
18	15, 16, 17	3eqtr4i 2793	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐵)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Ⅎwnfc 2907  {crab 3412  Vcvv 3450  {copab 5166   ↦ cmpt 5185

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-opab 5167  df-mpt 5186

This theorem is used by:  limsupequzmpt2  46821  liminfequzmpt2  46894