Theorem mptssid 46196

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 3471	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → 𝐵 ∈ V)
2	1	anim2i 629	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V))
3		rabid 3433	. . . . . . 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 2872	. . . . 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 46075	. . . . . . 7 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ 𝐴
11	5, 10	eqsstri 3977	. . . . . 6 ⊢ 𝐶 ⊆ 𝐴
12	11	sseli 3927	. . . . 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 5172	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵)}
16		df-mpt 5187	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
17		df-mpt 5187	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵)}
18	15, 16, 17	3eqtr4i 2794	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐵)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Ⅎwnfc 2908  {crab 3413  Vcvv 3451  {copab 5167   ↦ cmpt 5186

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rab 3414  df-v 3453  df-ss 3916  df-opab 5168  df-mpt 5187

This theorem is used by:  limsupequzmpt2  46672  liminfequzmpt2  46745