Theorem mptssid 45693

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 3451	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → 𝐵 ∈ V)
2	1	anim2i 623	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V))
3		rabid 3412	. . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V))
4	2, 3	sylibr 235	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V})
5		mptssid.2	. . . . . 6 ⊢ 𝐶 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
6	4, 5	eleqtrrdi 2850	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝐶)
7		simpr 485	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
8	6, 7	jca 516	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵))
9		mptssid.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
10	9	ssrab2f 45572	. . . . . . 7 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ 𝐴
11	5, 10	eqsstri 3961	. . . . . 6 ⊢ 𝐶 ⊆ 𝐴
12	11	sseli 3911	. . . . 5 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴)
13	12	anim1i 621	. . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
14	8, 13	impbii 210	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵))
15	14	opabbii 5140	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵)}
16		df-mpt 5155	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
17		df-mpt 5155	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵)}
18	15, 16, 17	3eqtr4i 2772	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐵)

Syntax hints:   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Ⅎwnfc 2886  {crab 3391  Vcvv 3431  {copab 5135   ↦ cmpt 5154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rab 3392  df-v 3433  df-ss 3900  df-opab 5136  df-mpt 5155

This theorem is referenced by:  limsupequzmpt2  46169  liminfequzmpt2  46242